lemieux-etal imex
2
A second-order accurate in time IMplicit-EXplicit
(IMEX) integration scheme for sea ice dynamics
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Jean-François Lemieuxa,⇤, Dana A. Knollb , Martin Loschc , Claude Girardd
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a
Recherche en Prévision Numérique environnementale/Environnement Canada, 2121
route Transcanadienne, Dorval, Qc H9P 1J3, Canada
b
Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, USA
c
Alfred-Wegener-Institut, Helmholtz-Zentrum für Polar-und Meeresforschung, Postfach
120161, 27515, Germany
d
Recherche en Prévision Numérique atmosphérique/Environnement Canada, 2121 route
Transcanadienne, Dorval, Qc H9P 1J3, Canada
Abstract
Current sea ice models use numerical schemes based on a splitting in time
between the momentum and continuity equations. Because the ice strength
is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence,
noise develops in the numerical solution and these models can even become
numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit-EXplicit (IMEX) time integration method. This
IMEX method was developed in the framework of an already implemented
Jacobian-free Newton-Krylov solver. The basic idea of this IMEX approach
is to move the explicit calculation of the sea ice thickness and concentration
inside the Newton loop such that these tracers evolve during the implicit
integration. To obtain second-order accuracy in time, we have also modified
the explicit time integration to a second-order Runge-Kutta approach and
by introducing a second-order backward di↵erence method for the implicit
integration of the momentum equation. These modifications to the code are
minor and straightforward. By comparing results with a reference solution
obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain
the same accuracy as the splitting in time but by using a time step that is
10 times larger. Results show that the second-order scheme is more than five
times more computationally efficient than the splitting in time approach for
an equivalent level of error.
⇤
Corresponding
author
Preprint
submitted to
Journal of computational physics
January 8, 2014
Email address: [email protected] (Jean-François Lemieux)
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Keywords: sea ice, IMEX method, backward di↵erence, Newton-Krylov
method, numerical accuracy
1. Introduction
Various mechanisms associated with sea ice dynamics play a key role in
shaping the ice cover of the polar oceans. To properly model the processes
of lead and pressure ridge formation, sea ice models require a sophisticated
representation of sea ice rheology, i.e. the relation between internal stresses,
material properties (ice strength) and deformations of the ice cover. Most
current sea ice models use the Viscous-Plastic (VP) formulation of Hibler
[1] to represent these ice interactions. The VP formulation leads to a very
nonlinear problem which is known to be difficult to solve.
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To the best of our knowledge, all sea ice model time integration schemes
are based on a splitting in time between the momentum and the continuity
equations (e.g., [1, 2, 3, 4, 5]). This means that when solving the momentum
equation, the thickness distribution (including the amount of open water) is
held constant at the previous time level (it, however, varies spatially). Once
the velocity field is obtained, the thickness distribution is advanced to the
next time level. Furthermore, an operator splitting approach is generally
used to separate the change of the thickness distribution associated with
advection and the growth/melt related to thermodynamic processes (e.g.,
[2, 3]). This paper focuses on dynamics and we therefore only discuss the
solution of the momentum equation and of the continuity equation without
the thermodynamic source terms.
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Current sea ice model numerical schemes su↵er from significant numerical
issues. First, as explained by Lipscomb et al. [2], the splitting in time approach leads to noise in the numerical solution and can even make the model
numerically unstable. As an illustrative example, consider ice converging
toward a coast due to an onshore wind; a stress gradient, associated with
an ice strength gradient, develops to oppose the wind stress. When using a
large time step with the splitting in time approach, an unrealistically large
ice strength gradient can occur. The stress gradient force can then overcompensate the wind stress and cause an unrealistic reversal of the flow (the ice
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then diverges at the coast). This instability, fundamentally numerical, can
be cured by reducing the time step. Unfortunately, this obviously increases
the total computational time. Lipscomb et al. [2] proposed a modification to
the ridging scheme in order to mitigate this problem.
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A second numerical issue is related to the solution of the momentum
equation. The rheology term, which determines the deformations of the ice
cover based on the internal ice stresses, causes the momentum equation to
be very nonlinear. Indeed, the VP rheology leads to a large change in the
internal stresses when going from a slightly convergent flow to a slightly divergent one (same idea for shear stresses). The current numerical solvers for
the momentum equation, however, have difficulties in finding the solution of
this very nonlinear problem. There are two main classes of schemes to solve
the momentum equation: the implicit solvers, which involve an outer loop
iteration (sometimes referred to as Picard iteration, [5, 6, 7]) and the ones
based on the explicit solution of the momentum equation using the ElasticVP approach [8, 9]. Both of these approaches, however, lead to a very slow
convergence rate [7, 9] if they converge at all [9, 10]. Because of this slow convergence rate, it is typical to perform a small number of Picard iterations or
of subcycling iterations. The approximate solution therefore contains residual errors which are carried on in the time integration.
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To resolve this slow convergence rate issue, Lemieux et al. [4] developed
a Jacobian-free Newton-Krylov (JFNK) implicit solver. They showed that
the JFNK solver leads to a more accurate solution than the EVP solver [10]
and that it is significantly more computationally efficient than a Picard approach [4]. Following the work of Lemieux et al. [4], Losch et al. [11] have
recently developed a parallel JFNK solver for the MIT general circulation
model with sea ice [12]. The numerical approaches of Lemieux et al. [4] and
Losch et al. [11], however, still rely on the splitting in time scheme and are
therefore susceptible to exhibit the numerical instability issue.
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It is the purpose of this paper to introduce a fast and accurate time integration scheme that resolves the instability associated with the splitting
in time approach. One possibility would be to solve fully implicitly the momentum and continuity equations. This avenue would imply significant modifications to the code and would be quite complex to implement. Instead,
the splitting in time issue is cured by using an iterated IMplicit-EXplicit
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(IMEX) approach when solving the momentum and continuity equations.
This approach is built around our existing JFNK solver. Basically, the idea
is to move the explicit calculation of the thickness distribution inside the
implicit Newton loop. We take this approach one step further by modifying
the time integration in order to get second-order accuracy in time for the full
system. To do so, we introduce a second-order Runge-Kutta scheme for the
advection operation and discretize in time the momentum equation using a
second-order backward di↵erence (as in [13]). This paper is inspired by the
work of [14, 15] on an iterated IMEX method for radiation hydrodynamics
problems.
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The main contribution of this paper is the development and demonstration of a first-of-a-kind second-order accurate in time iterated IMEX integration scheme for sea ice dynamics. This manuscript also shows the gain
in accuracy and computational time of the second-order IMEX method compared to the common first-order integration scheme based on the splitting in
time.
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It is worth mentioning that some authors have recently questioned the
validity of the VP rheology. Sea ice models based on a VP rheology do not
capture the largest deformations events [16] and statistics of simulated deformations do no match observations [16] in both space and time [17]. While
some authors propose new and very di↵erent formulations of ice interactions
[18, 19], others claim that a VP rheology with modified yield curve and flow
rule can adequately represent the sea ice deformations [20]. These new physical parameterizations, under evaluation, also lead to very nonlinear problems
which would also clearly benefit from the availability of reliable and efficient
numerical schemes.
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This paper is structured as follows. Section 2 describes the sea ice momentum equation with a VP formulation and the continuity equation. In
section 3, the discretization of the momentum and continuity equations and
the descriptions of the standard splitting in time and new IMEX integration
schemes are presented. In section 4, more information about the model is
given. The description of the experiments and the results are outlined in
section 5. A discussion and concluding remarks are provided in section 6.
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2. Sea ice momentum and continuity equations
As the ratio between the horizontal and the vertical scales is O(1000
km/10 m) = O(105 ), sea ice dynamics is often considered to be a twodimensional problem [21]. The two-dimensional sea ice momentum equation
is obtained by integrating in the vertical the momentum equation. It is given
by
Du2
= ⇢hf k ⇥ u2 + ⌧a ⌧w + r ·
⇢hgrHd ,
(1)
Dt
where ⇢ is the density of the ice, h is the ice volume per unit area (or the
D
mean thickness and just referred to as thickness in this paper), Dt
is the
total derivative, f the Coriolis parameter, u2 = ui + vj the horizontal sea ice
velocity vector, i, j and k are unit vectors aligned with the x, y and z axis
of our Cartesian coordinates, ⌧a is the wind stress, ⌧w the water stress, the
internal ice stress tensor (r · is defined as the rheology term), g the gravity
and Hd the sea surface height. The subscript in u2 indicates that it is a 2-D
vector and it is used to distinguish u2 from the vector u obtained from the
spatial discretization (explained in section 3).
⇢h
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As in Tremblay and Mysak [3], the sea surface tilt is expressed in terms of
the geostrophic ocean current. Using a quadratic law and constant turning
angles ✓a and ✓w , ⌧a and ⌧w are expressed as [22]
⌧a = ⇢a Cda |uga |(uga cos ✓a + k ⇥ uga sin ✓a ),
⌧w = ⇢w Cdw |u2
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ugw |[(u2
ugw ) cos ✓w + k ⇥ (u2
ugw ) sin ✓w ],
(2)
(3)
where ⇢a and ⇢w are the air and water densities, Cda and Cdw are the air and
water drag coefficients, and uga and ugw are the geostrophic wind and ocean
current. As u2 is much smaller than uga , it is neglected in the expression for
the wind stress.
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The VP constitutive law, that relates the internal stresses and the strain
rates, can be written as [1]
ij
= 2⌘ ✏˙ij + [⇣
⌘]✏˙kk
ij
5
P
ij /2,
i, j = 1, 2,
(4)
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where ij are the components of the ice stress tensor, ij is the Kronecker
@v
delta, ✏˙ij are the strain rates defined by ✏˙11 = @u
, ✏˙22 = @y
and ✏˙12 =
@x
1 @u
@v
( + @x ), ✏˙kk = ✏˙11 + ✏˙22 , ⇣ is the bulk viscosity, ⌘ is the shear viscos2 @y
ity and P is a pressure-like term which is a function of the ice strength.
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With a two-thickness category model, the ice strength Pp is parameterized
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as
Pp = P ⇤ h exp[ C(1
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A)],
(5)
where P ⇤ is the ice strength parameter, A is the sea ice concentration and C
is the ice concentration parameter, an empirical constant characterizing the
strong dependence of the compressive strength on sea ice concentration [1].
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The formulation of the bulk and shear viscosities depends on the yield
curve and the flow rule. In the following, the elliptical yield curve with a
normal flow rule [1] is used. In this case, the bulk and shear viscosities are
given by
⇣=
Pp
,
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(6)
⌘ = ⇣e 2 ,
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(7)
1
2
where 4 = [(✏˙211 + ✏˙222 )(1 + e 2 ) + 4e 2 ✏˙212 + 2✏˙11 ✏˙22 (1 e 2 )] , and e is the
aspect ratio of the ellipse, i.e. the ratio of the long and short axes of the
elliptical yield curve.
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When 4 tends toward zero, equations (6) and (7) become singular. To
avoid this problem, ⇣ is capped using an hyperbolic tangent [7]
⇣ = ⇣max tanh(
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Pp
).
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(8)
As in equation (7), ⌘ = ⇣e 2 . The coefficient ⇣max is set to the value
proposed by Hibler [1]: 2.5 ⇥ 108 Pp (this is equivalent to limiting 4 to a
minimum value of 2 ⇥ 10 9 s 1 ). As opposed to the regularization introduced
by Hibler [1], this formulation for ⇣ is continuously di↵erentiable.
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We use a replacement closure similar to the one presented in Kreyscher
et al. [23]. The pressure term is given by
P = 2⇣4.
The continuity equations for the thickness and the concentration are given
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(9)
by
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@h
+ r · (u2 h) = Sh ,
@t
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(10)
@A
+ r · (u2 A) = SA ,
(11)
@t
where Sh and SA are thermodynamic source terms. Note that A is limited
above to 1.0. This does not a↵ect the conservation of mass as the mass per
m2 is given by ⇢h. The source terms in equations (10) and (11) are set to zero
in the simulations for this paper (unless otherwise stated) as we concentrate
on matters related to the dynamics.
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3. Numerical approaches
3.1. Temporal discretization
The advection of momentum is neglected as it is small compared to the
other terms in the momentum equation (as done in [6, 8]). The momentum
and continuity equations are solved at time levels t, 2 t, 3 t, . . . where
t is the time step and the index n = 1, 2, 3, . . . refers to these time levels.
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The standard numerical approach involves a Splitting In Time (SIT) between the implicit momentum and explicit continuity equations. This splitting implies that h and A (and therefore Pp ) are considered to be known in
the momentum equation as they are held at the previous time level. Using a
backward Euler approach for the acceleration term, the u and v momentum
equations at time level n are written as
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⇢hn
1 (u
n
un 1 )
n
= ⇢hn 1 f v n + ⌧au
t
n
⌧wu
+
7
@
n
n
11 (Pp
@x
1
)
+
@
n
n
12 (Pp
@y
1
)
, (12)
⇢h
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n 1 (v
n
vn 1)
=
t
⇢h
n 1
fu
n
n
+ ⌧av
n
⌧wv
+
@
n
n
22 (Pp
1
)
@y
+
@
n
n
12 (Pp
1
)
@x
, (13)
where the sea surface tilt term is ignored here to simplify the presentation.
As the water drag and the rheology term are written in terms of the velocity
field, the only unknowns in equations (12) and (13) are un and v n . Once these
equations are solved for un and v n everywhere on the grid, the thickness and
concentration fields are advanced in time according to
(hn
hn 1 )
+ r · (un2 hn 1 ) = 0,
t
(14)
(An
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An 1 )
+ r · (un2 An 1 ) = 0,
(15)
t
for which we use a first-order (in space) upstream scheme (as in [3, 23, 24]).
We introduce the operator L given by
hn = L(hn 1 , un2 ),
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(16)
which allows one to write concisely the explicit calculation of hn based on the
upstream scheme (same idea for An ). This scheme is stable if the CourantFriedrichs-Lewy (CFL) condition max(u, v) < xt is respected, with x being
the spatial resolution.
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This scheme for the integration of the momentum and continuity equations is first-order accurate in time as a consequence of the first-order treatment in both the momentum and continuity equations, and as a result of the
SIT splitting error which is not iterated. We here introduce a few straightforward modifications that allows one to solve simultaneously these equations
with second-order accuracy in time.
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First, we introduce a second-order backward di↵erence (BDF2, [13]) approach for the momentum equation. Hence, the u and v equations are written
as
⇢hn 3 n
( u
t 2
2un
1
1
n
+ un 2 ) = ⇢hn f v n + ⌧au
2
8
n
⌧wu
+
@
n
n
11 (Pp )
@x
+
@
n
n
12 (Pp )
@y
, (17)
⇢hn 3 n
( v 2v n
t 2
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1
1
+ vn 2) =
2
n
n
⇢hn f un +⌧av
⌧wv
+
@
n
n
22 (Pp )
@y
+
@
n
n
12 (Pp )
@x
, (18)
where h, A and Pp are at time level n because BDF2 is used along with
IMEX (as explained below).
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We note in passing that a second-order Crank-Nicolson scheme for the
momentum equation was not successful because the water stress term leads
to an an undamped oscillation. For more details, the reader is referred to
Appendix A.
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Secondly, to obtain second-order accuracy in time for the continuity equations, we use a second-order Runge-Kutta (RK2) predictor-corrector approach to obtain hn and An . Hence, they are obtained in two steps by
doing
(h⇤
(hn
n
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hn 1 )
+ r · (un2 1 hn 1 ) = 0,
t/2
hn 1 )
n
+ r · (u2
t
1
2
h⇤ ) = 0,
(19)
(20)
1
where u2 2 = (un2 1 + un2 )/2. h⇤ is centered in time as t/2 is used to
perform the advection for the predictor step. Both steps use the upstream
scheme. We introduce the operator hn = LRK2 (hn 1 , un2 1 , un2 ), similar to
the one in equation (16), in order to denote the two-step calculation of hn .
The RK2 approach with the upstream scheme has the same CFL condition
than the first-order scheme.
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Before we introduce our third modification and explain how these equations can be solved simultaneously for un , v n , hn and An , we need to present
the JFNK solver.
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3.2. Spatial discretization and boundary conditions
The components of the velocity (u and v) are positioned on the Arakawa
C-grid. A Dirichlet boundary condition is applied at an ocean-land boundary (u = 0, v = 0) and a Neumann condition at an open boundary (i.e.,
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the spatial derivatives of the components of velocity in the normal direction with the open boundary are chosen to be zero). Gradients of h and
A are also set to zero at an open boundary. For stability, the ice strength
Pp is set to zero at the open boundaries [25]. A f-plane approximation is
used with f = 1.46 ⇥ 10 4 s 1 . Spatial derivatives (in the rheology term) are
discretized using centered finite di↵erences except close to land boundaries
where second order accurate Taylor series expansions are used. As opposed
to our work in [4] and [10], the viscous coefficients are calculated following
the method described in Bouillon et al. [9]. The spatial discretization (with
nx tracer points in one direction and ny in the other one) leads to a system
of N = (ny(nx + 1) + nx(ny + 1)) nonlinear equations for the velocity components and (nx + 2)(ny + 2) equations for each h and A (this includes the
boundary conditions).
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3.3. The JFNK solver
We give a brief overview of the JFNK implementation. More details can
be found in [4, 10, 26]. The u and v equations to be solved at time level n
for each grid cell can be written as
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⇢hlu
(↵un + un
t
1
+ u
n 2
)=
⇢hlv
(↵v n + v n 1 + v n 2 ) =
t
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n
n
⇢hlu f vavg
+⌧au
n
n
@ 11
(Ppl )
@ 12
(Ppl )
n
⌧wu +
+
,
n
n
⇢hlv f unavg +⌧av
⌧wv
+
@x
@
n
l
22 (Pp )
@y
@y
+
@
n
l
12 (Pp )
@x
(21)
, (22)
where hu is the thickness evaluated at the u location on the C-grid and vavg
is the average of the four v components surrounding the u location (similar
idea for hv and uavg ). The parameters ↵, and are respectively equal to
1, -1 and 0 for the SIT approach and to 23 , -2 and 12 for the BDF2 scheme.
The superscript l is n 1 for the SIT method while it is n with the IMEX
method (explained below).
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From both approaches, we obtain equations that are functions of un and
v n . The spatial discretization of equations (21) and (22) leads to a system
of N nonlinear equations with N unknowns that can be concisely written as
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Am (un )un = b(un ),
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(23)
where Am is an N ⇥ N matrix. We added a subscript m to distinguish the
system matrix from the ice concentration vector A. The vector un , of size N ,
is formed by stacking first the u components followed by the v components.
The vector b is a function of the velocity vector un because of the water
stress term. Note that the system of equations also depends on the vectors
hn and An for IMEX and on hn 1 and An 1 when using the SIT approach.
The systems of equations to be solved are di↵erent whether the SIT or BDF2
approach is used (the two methods lead to di↵erent system matrix, vector b
and solution). We drop the superscript n knowing that we wish to find the
solution u = un . We introduce the residual vector F(u):
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F(u) = Am (u)u
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b(u).
(24)
The residual vector F(u) is useful as it allows one to evaluate the quality
of the approximate solution as F(u) = 0 if the solution is fully converged.
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The Newton method is used to solve the nonlinear system of equations
given in (23). The iterates obtained during the Newton method are referred
to as uk where the superscript k corresponds to the Newton iteration number.
This nonlinear method is based on a multivariate Taylor expansion around
a previous iterate uk 1 :
F(uk
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1
0
+ uk ) ⇡ F(uk 1 ) + F (uk 1 ) uk .
The higher order terms are neglected in the expression above. Setting
F(uk 1 + uk ) = 0, uk = uk uk 1 can be obtained by solving the linear
system of N equations:
J(uk 1 ) uk =
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(25)
0
F(uk 1 ),
(26)
where the system matrix J ⌘ F is the Jacobian, an N ⇥ N matrix whose
entries are Jqr = @Fq (uk 1 )/@(ukr 1 ) (where q = 1, N and r = 1, N ). For
k = 1, an initial iterate u0 needs to be provided. The initial iterate here is
the previous time level solution un 1 . Once the linear system of equations
(26) is solved, the next iterate is given by
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⇥
⇤
1
uk = uk
1
+
uk ,
(27)
where = 1, 12 , 14 , 8 is iteratively reduced until ||F(uk )|| < ||F(uk 1 )|| or
until = 18 . The symbol || || denotes the L2-norm. This linesearch approach
is an addition compared to the previous model versions described in Lemieux
et al. [4] and Lemieux et al. [10] (see also Losch et al. [11]). This method
greatly improves the robustness of the nonlinear solver.
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The linear system of equations in (26) is solved using the Flexible Generalized Minimum RESidual (FGMRES, [27]) method. Krylov methods such
as FGMRES approximates the solution in a subspace of small dimension.
When creating the subspace, Krylov methods only need the product of J
times certain vectors (see Knoll and Keyes [28] for details). The Jacobian
matrix therefore does not need to be formed per se but only its action on
a vector is required. Given a certain vector w formed during the Krylov
process, the product of J times w can be approximated by
J(uk 1 )w ⇠
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F(uk
1
+ ✏w)
✏
F(uk 1 )
,
(28)
where ✏ is a small perturbation.
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To speed up convergence of the linear solution, the system of equations
is transformed using right preconditioning. The preconditioning operator is
based on the matrix Am linearized with the previous iterate and involves 10
iterations of a Line Successive Over Relaxation (LSOR) scheme [4, 26]. The
preconditioning operator is slightly di↵erent whether the SIT or the BDF2
method is used. This is a consequence of the di↵erent formulation of the
inertial term which just leads to a multiplying factor of 32 for BDF2 and of 1
for SIT.
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To improve robustness and computational efficiency, an inexact Newton
method [29] is employed. With this approach, a loose tolerance is used in
early Newton iterations and it is progressively tighten up as the nonlinear
solution is approached. The preconditioned FGMRES method solves the
linear system of equations until the linear residual is smaller than (k) k
F(uk 1 ) k where (k) is the tolerance of the linear solver at iteration k (a
value smaller than 1). The tolerance of the linear solver with this inexact
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Newton approach is given by
(
h ini ,
(k) =
||F(uk
||F(uk
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1 )||
2 )||
i↵
if ||F(uk
, if ||F(uk
1
)||
1
)|| < r.
r,
(29)
The tolerance ini for the initial stage is set to 0.99. The exponent ↵ is
set to 1.5 and r = 23 ||F(u0 )||. Because of the linesearch approach, a more
aggressive evolution of the linear tolerance is used compared to the settings
in [4, 10]. The tolerance (k) is also forced to be larger than 0.1 to prevent
excessive use of the linear solver which tends to slow down the nonlinear
solver. We will get back to this issue later in the paper.
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Finally, a termination criterion (defined by nl ) for solving the nonlinear
system of equations is also needed. The JFNK solver stops iterating after the
L2-norm of the residual is lower than nl ||F(u0 )||. JFNK fails to converge
when the termination criterion is not reached in kmax =100 iterations.
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The JFNK algorithm with the SIT approach and the first-order upstream
scheme is:
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1. Start with an initial iterate u0
do k = 1, kmax
2. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES
3. uk = uk 1 + uk
4. If ||F(uk )|| < nl ||F(u0 )|| stop
enddo
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5. Calc hn = L(hn 1 , un ) and An = L(An 1 , un )
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where the initial iterate u0 is the previous time level solution and uk = un
once it has converged. The matrix J and the vector F are functions of h and
A at the previous time level, i.e. hn 1 and An 1 (note that SIT is technically
an IMEX method, but it is not iterated).
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The iterated IMEX approach (simply referred to as IMEX) now allows
one to solve for un , vn , hn and An simultaneously. In order to do this, the
explicit calculations of the thickness and concentration are moved inside the
Newton loop.
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1. Start with an initial iterate u0
do k = 1, kmax
2. Calc hk = L(hn 1 , uk 1 ) and Ak = L(An 1 , uk 1 )
3. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES
4. uk = uk 1 + uk
5. If ||F(uk )|| < nl ||F(u0 )|| stop
enddo
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where in this case J and F are function of hk and Ak .
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To obtain second-order accuracy in time, the latter algorithm can be modified by using the LRK2 advection operator and by using the BDF2 method.
Hence, the BDF2-IMEX-RK2 algorithm is given by
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1. Start with an initial iterate u0
do k = 1, kmax
2. Calc hk = LRK2 (hn 1 , un 1 , uk 1 ) and Ak = LRK2 (An 1 , un 1 , uk 1 )
3. ‘‘Solve’’ J(uk 1 ) uk = F(uk 1 ) with FGMRES
4. uk = uk 1 + uk
5. If ||F(uk )|| < nl ||F(u0 )|| stop
enddo
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To ensure fast nonlinear convergence in the context of the IMEX or
BDF2-IMEX-RK2 scheme, it is crucial to take into account the change in
h and A associated with a change of velocity in the evaluation of J times
a certain Krylov vector w (equation (28)). Hence, with the BDF2-IMEXRK2 scheme, F(uk 1 + ✏w) is a function of h+ = LRK2 (hn 1 , un 1 , u+ ) and
A+ = LRK2 (An 1 , un 1 , u+ ) where u+ is uk 1 + ✏w (same idea for IMEX by
using the simpler operator L).
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For simplicity, the same notation is used for the three algorithms given
above. However, as they do not solve the same nonlinear systems of equations, they lead to di↵erent Jacobian matrices, residual vectors and solutions.
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A truncation error analysis, that demonstrates second-order accuracy in
time for BDF2-IMEX-RK2, is given in Appendix B.
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14
Symbol
⇢
⇢a
⇢w
Cda
Cdw
✓da
✓dw
f
P*
C
e
Definition
sea ice density
air density
water density
air drag coefficient
water drag coefficient
air turning angle
water turning angle
Coriolis parameter
ice strength parameter
ice concentration parameter
ellipse ratio
value
900 kg m 3
1.3 kg m 3
1026 kg m 3
1.2 ⇥ 10 3
5.5 ⇥ 10 3
25
25
1.46 ⇥ 10 4 s 1
27.5⇥103 N m
20
2
2
Table 1: Physical parameters for the numerical simulations
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4. Information about the model
Our pan-Arctic regional model can be run at four possible spatial resolutions: 10, 20, 40 and 80 km (square cartesian grids). The model uses two
thickness categories and a zero-layer thermodynamics (described in [3]). The
sea ice model is coupled thermodynamically to a slab ocean model. Climatological ocean currents are used to force the sea ice model and to advect heat
in the ocean. The wind stress is calculated using the geostrophic winds derived from the National Centers for Environmental Prediction and National
Center for Atmospheric Research (NCEP/NCAR) six hour reanalysis of sea
level pressure [30].
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Tables (1) lists the values of the physical parameters used for the simulations in this paper.
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For all the 2-D experiments, we use revision 317 of our model with small
modifications to perform the experiments described below. The code is serial.
All runs were performed on a machine with 2 Intel E5520 quad-core CPU
at 2.26 GHz with 8 MB of cache and 72 GB of RAM. The compiler is GNU
fortran (GCC) 4.1.2 20080704 (Red Hat 4.1.2-54), 64 bits. The optimization
option O3-↵ast-math was used for all the runs.
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To introduce and better illustrate the SIT instability, a few 1-D experiments are performed. Revision 89 of our 1-D model is used for all the 1-D
experiments. A detailed description of the sea ice dynamic equations in 1-D
can be found in [2].
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5. Results
A series of one day numerical experiments in 1-D and 2-D are performed
for the di↵erent time integration schemes at spatial resolutions of 40 and 20
km. The base set of numerical experiments use the SIT algorithm (referred
to as SIT). The second set of numerical experiments use the iterated IMEX
algorithm (referred to as IMEX). The final set of numerical experiments use
the BDF2 scheme along with IMEX and the RK2 advection scheme (referred to as BDF2-IMEX-RK2). For each series, one day experiments are
performed with di↵erent time steps ( t). To ensure that the CFL condition is respected, the maximum t at 40-km resolution is set to 360 min
while it is 180 min for a resolution of 20 km (At these resolutions and maximum time steps, the CFL criterion is not violated for ice velocities 1 ms 1 ).
It was observed that the solver had difficulties at the beginning of the
time integration (with small wind and ice starting from rest). A value of
✏ = 10 7 , in the evaluation of the Jacobian times a vector (equation (28)),
improves robustness compared to the value of 10 6 used in [4, 10]. Robustness was improved for the first few time levels by setting ✏ = 10 8 instead
of 10 7 when the Newton iteration is larger than 50. This robustness issue
is not a major problem as it has not been observed in realistic experiments.
It is possible that a more sophisticated way of choosing ✏ (as described in
[28]) or an exact Jacobian-times-vector operation by automatic di↵erentiation [11] could improve robustness for these idealized experiments, but this
is not explored in this paper. As these few initial time levels are not representative of the usual behavior of the solver, only the last 12 hours of the one
day integration are used to compute metrics to compare the di↵erent time
integration approaches.
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5.1. 1-D experiments
For these 1-D experiments, the domain is 2000 km long with solid walls
at both ends. There is a no inflow/outflow condition at the walls: i.e., the
velocity is zero. The spatial resolution is 20 km. The initial thickness field
is 1 m everywhere and the sea ice concentration is 0.95. The ice starts from
rest. The westerly wind is zero at the beginning and is increased smoothy
according to uga (t) = (1 e t/⌧ )ug⇤
a with ⌧ , a time constant set to 6 hours,
1
and |ug⇤
|
=10
m
s
being
the
same
everywhere.
a
To assess the quality of these approximate solutions, a 24-h reference solution is obtained by using a time step of 1 s (with BDF2-IMEX-RK2). We
then compare the 24-h sea ice thickness field obtained with an integration
scheme using a certain t with the reference solution. Thickness is used
because it acts as an integrator of all the errors produced during the time
integration. The Root Mean Square Error (RMSE) between a thickness field
and the reference thickness field is calculated for all the experiments. The
RMSE should decrease with t for all three series of experiments. BDF2IMEX-RK2 should be the most accurate and lead to second-order accuracy
in time while the other two series (SIT and IMEX) are expected to be firstorder accurate in time. The termination criterion is nl = 10 6 for all the
experiments.
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Fig. 1a indeed confirms that SIT and IMEX are both first order accurate
in time (the slope is ⇠1 on a log-log plot). This figure shows the RMSE
between an approximate solution (thickness) and the reference solution as a
function of the time step. Despite some wiggling, BDF2-IMEX-RK2 exhibits
second-order accuracy in time. For any t, the BDF2-IMEX-RK2 solution
is more than one order of magnitude more accurate than the IMEX and SIT
ones. The improvement of IMEX over SIT is small except for large t. This
implies that for smaller t, the splitting errors are smaller than the standard
first-order discretization errors. The sudden increase in the RMSE for SIT
for t larger than 60 min is due to noise in the thickness field near both walls.
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The fact that the approximate solution for SIT is contaminated by noise
makes it more difficult for JFNK to obtain the velocity field solution. This is
illustrated in Fig. 1b. Whereas both IMEX and BDF2-IMEX-RK2 need less
than 20 Newton iterations (on average), SIT behaves di↵erently than these
two schemes for t larger than 15 min. Indeed, the mean number of Newton
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iterations for SIT increases significantly for
failure of JFNK for t = 120 min.
t > 15 min. There was even a
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These additional Newton iterations for SIT have an impact on the total CPU time as can be seen in Fig. 1c. While SIT is more efficient than
IMEX and BDF2-IMEX-RK2 for small t, the additional Newton iterations
for t > 15 min causes SIT to be more costly. Hence, BDF2-IMEX-RK2
is always significantly more accurate than SIT and it is also more computationally efficient than SIT for typical time steps (e.g. t = 60 min).
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Fig. 2 displays how the errors are spatially distributed. The reference
thickness and velocity solutions are respectively shown on Fig. 2a and 2b.
The ice has piled up and the velocity exhibits strong convergence at the wall.
The ice concentration has reached 1.0 close to the wall (not shown).
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The di↵erence between the thickness obtained with SIT when using a time
step of 120 min or 180 min and the reference solution are respectively shown
on Fig. 2c and Fig. 2d in black. Similar to the results of Lipscomb et al. [2],
there is noise in the approximate solution in the region of convergence. It is
also observed that errors are also present on the western side of the domain
where the ice is diverging. The error is, however, more localized than close
to the eastern wall. The maximum errors are respectively 2.5 cm and 8.1
cm for t of 120 and 180 min. These figures also demonstrate that the
noise is notably smaller everywhere on the domain with BDF2-IMEX-RK2
(in blue). In this case, the maximum errors are 0.1 cm ( t = 120 min) and
0.32 cm ( t = 180 min). As opposed to the SIT scheme, the IMEX approach
decreases the errors close to the eastern wall but does not significantly a↵ect
the noise on the other side of the domain where the ice diverges (not shown).
5.2. 2-D experiments
Experiments in 2-D are performed at 40 and 20-km resolutions. The
initial conditions for these one day are the same than in [10]. These experiments are performed starting on 17 January 2002 00Z. As in Lemieux et al.
[10], this 24-hour period was chosen because it is characterized by typical
conditions in the Arctic: a high pressure system close to the Beaufort Sea,
convergence north of Greenland and ice flowing south through Fram Strait.
The thermodynamics and the ocean currents are set to zero for these idealized
experiments. The ice starts from rest. It is then accelerated by a smoothly
18
(a)
100
−3
10
−4
RMSE (m)
10
(b)
90
Mean nb of Newton iterations
−2
10
SIT
IMEX
BDF2−IMEX−RK2
80
70
60
50
40
30
20
10
0
−5
10
0
20
40
60
80
100
120
140
160
180
Time step (min)
1
10
(c)
−6
SIT
IMEX
BDF2−IMEX−RK2
Total CPU time (s)
10
−7
10
−8
10
SIT
IMEX
BDF2−IMEX−RK2
0
10
1
2
10
10
Time step (min)
0
10
−1
10
0
3
10
20
40
60
80
100
120
140
160
180
Time step (min)
Figure 1: RMSE (a), mean number of Newton iterations per time level (b) and total CPU
time (c) as a function of the time step. The mean number of Newton iterations and total
CPU time were calculated for the last 12 h of the integration. Black curve with triangles
is for the SIT scheme, red curve with diamonds is for IMEX while the blue curve with
circles is BDF2-IMEX-RK2. This is a 1-D experiment with a spatial resolution of 20 km.
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increased wind stress field. The geostrophic wind field on 18 January 2002
00Z is used but it is ramped up according to
uga (t) = (1
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e
t/⌧
)ug⇤
a ,
(30)
where ug⇤
a is the geostrophic wind field on 18 January 2002 00Z, t is the time
(starting on 17 January 2002 00Z) and ⌧ is set to 6 hours as in the 1-D
experiments.
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A reference solution is again obtained by using a time step of 1 s (with
BDF2-IMEX-RK2). We then compare the sea ice thickness field obtained on
18 January 2002 00Z with the reference solution valid at the same time. As
in the 1-D experiments, the termination criterion is set to nl = 10 6 .
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Fig. 3a shows the 20-km reference solution concentration field on 18 January 2002 00Z while Fig. 3b displays the reference solution velocity field at
19
1.2
0.2
(a)
(b)
1
0.15
u (m/s)
h (m)
0.8
0.6
0.1
0.4
0.05
0.2
0
0.1
t=1 s (Ref)
0
500
1000
1500
0
2000
0.1
(c)
SIT − Ref
BDF2−IMEX−RK2 − Ref
0
−0.05
−0.1
500
(d)
1000
1500
2000
SIT − Ref
BDF2−IMEX−RK2 − Ref
0.05
h (m)
h (m)
0.05
t=1 s (Ref)
0
0
−0.05
t=2 h
0
500
1000
1500
2000
Distance (km)
−0.1
t=3 h
0
500
1000
1500
2000
Distance (km)
Figure 2: 1-D reference solution ice thickness (a) and velocity (b) fields. Di↵erence between
the thickness field obtained with the SIT approach (in black) or with BDF2-IMEX-RK2
(in blue) and the reference solution for t = 120 min (c) and t = 180 min (d). The
spatial resolution is 20 km. The x-axis for these graphs is the distance in km from the
western wall.
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the same valid time. The reference thickness solution is shown in Fig. 7a.
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Fig. 4 shows, for the di↵erent schemes, the RMSE as a function of the
time step on a log-log plot for spatial resolutions of 40 km (a) and 20 km
(b). The RMSE is calculated only where the concentration of the reference
solution is above 50%. The behavior of the time integration scheme is qualitatively the same at both resolutions. We therefore concentrate on the 20-km
resolution results. The SIT and IMEX schemes lead to first-order accuracy
in time while BDF2-IMEX-RK2 clearly demonstrates that it is second-order
accurate in time over a wide range of t. There seems to be error saturation
for large t as a flattening of the curve is observed.
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As the continuity and momentum equations are solved simultaneously
20
(b)
(a)
Figure 3: Ice concentration (a) and velocity field (b) at 20-km resolution on 18 January
2002 00Z obtained with BDF2-IMEX-RK2 with a time step of 1 s. These 2-D fields
form the reference solution. For clarity, only one velocity vector out of 16 is shown. The
continents are in gray.
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with BDF2-IMEX-RK2, we verify that the scheme also leads to second-order
accuracy in time for the velocity field. Fig. 5 shows the RMS of the magnitude of the velocity error (referred to as RMSEv) between an approximate
solution and the reference solution as a function of t. This result demonstrates second-order accuracy in time for the velocity field when using the
BDF2-IMEX-RK2 scheme.
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Consistent with the findings of Lipscomb et al. [2], we observe that SIT
is less sensitive in 2-D than in 1-D. Shear stress tends to help the numerical
scheme. A test with an elliptical yield curve with a very large aspect ratio
of 1000 (i.e., with very small resistance to shear deformations) shows that
results in 2-D exhibit a similar behavior to results in 1-D (the mean number
of Newton iterations and RMSE for SIT increases significantly for large t,
not shown). Our results also suggest that our model is less sensitive to the
SIT instability than the one of Lipscomb et al. [2]. This is likely because we
use a two-thickness category model as opposed to their multi-category model.
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Fig. 6a and Fig. 6b respectively show the mean number of Newton iterations per time level (last 12 h) and the total CPU time required for the
last 12 h of the one day integration, as a function of t, for the di↵erent
time integration schemes. As opposed to the 1-D experiments, the number
of Newton iterations for SIT is about the same as for IMEX and BDF221
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IMEX-RK2 even for large t. BDF2-IMEX-RK2 requires roughly 10-25%
more total CPU time than SIT for the same t. As this is not due to an increase in the number of Newton iterations (the number is even slightly lower
for BDF2-IMEX-RK2), the extra CPU time for BDF2-IMEX-RK2 is rather
a consequence of the additional operations inside the Newton loop (the twostep advection operator). However, comparing the computational efficiency
of SIT and BDF2-IMEX-RK2 for the same t is not a fair comparison as
the integration schemes do not lead to the same accuracy. As an example,
BDF2-IMEX-RK2 with a t of 90 min leads to an approximate solution that
is more accurate (RMSE of 1.77 ⇥ 10 4 m) than the one obtained with SIT
with t= 10 min (RMSE of 2.86 ⇥ 10 4 m, Fig. 4b). As the total CPU time
required by BDF2-IMEX-RK2 with t= 90 min is 146 s and the one for SIT
with t= 10 min is 775 s, this means that the second-order scheme is more
than five times faster than the SIT integration scheme to obtain the same
accuracy.
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Fig. 7c shows how the thickness errors are spatially distributed on the
pan-Arctic domain when using BDF2-IMEX-RK2 with t=90 min. This
can be compared to the errors obtained with SIT for the same t of 90 min
(Fig. 7b). Fig. 7b shows that notable errors are found at many places in the
domain, with the largest errors close to the coast lines. The largest errors in
SIT with t=90 min is -7.6 cm while the maximum error is reduced to 0.34
cm with BDF2-IMEX-RK2 when using the same time step. As mentioned
earlier, SIT needs a t=10 min to obtain a comparable RMSE than the one
obtained with BDF2-IMEX-RK2 with t=90 min. The spatial errors for
SIT for a t of 10 min are shown on Fig. 7d. Qualitatively speaking, it can
be observed that the errors in Fig. 7c and Fig. 7d are of similar magnitude,
although the spatial patterns are di↵erent. The largest error for SIT with
t=10 min is -0.78 cm.
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5.3. Robustness
We have first assessed the robustness of the BDF2-IMEX-RK2 scheme
when using winds that change more abruptly. We repeated the 40 km resolution experiments of Section 5.2 but with winds that change a lot more
quickly. The time constant in equation (30), that determines how quickly
the winds are ramped up, was set to 1 hour (instead of 6 hours). Results
demonstrate that the BDF2-IMEX-RK2 scheme still leads to second-order
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accuracy in time (not shown).
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We have also investigated how robust is our JFNK solver when used in
the context of the BDF2-IMEX-RK2 scheme or in the context of the SIT
first-order approach. We ran the 2-D model for five years (2002-2007) at 40
and 20-km resolutions with either BDF2-IMEX-RK2 or SIT and counted the
number of failures of JFNK. For all these experiments, t is 30 min and
4
nl = 10 . Note that realistic wind forcing was used and thermodynamic
source terms were included (through operator splitting) for these long simulations.
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The introduction of the linesearch globalization and to a lesser extent of
the Bouillon et al. [9] approach for the calculation of the viscous coefficients
clearly improved the robustness of our JFNK solver when compared to the
first version described in [4]. For these five-year integrations, JFNK within
both the SIT and BDF2-IMEX-RK2 schemes did not fail at 40-km resolution.
However, at 20-km resolution, JFNK failed a few times for both integration
schemes. In terms of percentage, the failure rate is 0.027 % for SIT while it
is 0.025 % for BDF2-IMEX-RK2. Losch et al. [11] report a failure rate of
0.006% with a SIT approach over a 50 year simulations for a spatial resolution of 27 km.
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6. Discussion and concluding remarks
To our knowledge, we have demonstrated for the first time second-order
temporal accuracy in a sea ice dynamic model. This second-order scheme
was implemented relatively easily from a Splitting In Time (SIT) scheme using a Jacobian-free Newton-Krylov (JFNK) nonlinear solver. Basically, three
minor modifications were made to this configuration to get second-order accuracy in time. First, the advection operation was moved inside the Newton
loop such that the ice thickness and concentration fields are updated along
with the velocity field during the Newton iteration. Secondly, the first-order
explicit advection operation was upgraded to a second-order Runge-Kutta
(RK2) predictor-corrector approach. Finally, in order to get second-order
accuracy, the backward Euler time discretization in the momentum equation
was replaced by a second-order backward di↵erence formula (BDF2) integration scheme. We refer to this new iterated IMplicit-EXplicit (IMEX) scheme
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as BDF2-IMEX-RK2. This implementation is a lot more straightforward
than the development of a fully implicit scheme would have been.
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The Root Mean Square Error (RMSE) between thickness fields obtained
with di↵erent time steps ( t) and a reference solution thickness field demonstrates that BDF2-IMEX-RK2 is second-order accurate in time. The supporting analysis can be found in Appendix B. Results at 40 and 20-km resolutions lead qualitatively to the same conclusions. For the same t, BDF2IMEX-RK2 is always more than one order of magnitude more accurate than
the SIT approach. As an example, the approximate solution obtained with
BDF2-IMEX-RK2 with t= 90 min is more accurate than the one obtained
with SIT with t=10 min. Hence, to get the same level of accuracy than
SIT, significantly larger time steps can be used with BDF2-IMEX-RK2 which
leads to a decrease in the computational time. This efficiency gain is greater
than a factor of 5 at 20-km resolution.
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The implementation of this efficient second-order accurate in time scheme
was possible because our nonlinear solver for the momentum equation is a
Newton-Krylov scheme. As the EVP solver [8] is an explicit scheme, the
IMEX approach would not be possible with this method. On the other
hand, IMEX could be implemented in the framework of a Picard iteration
(e.g. [5, 6, 7]) although the Picard solver is known to exhibit a very inefficient
nonlinear convergence [7, 11].
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To maintain the fast nonlinear convergence of JFNK with the IMEX approach, it is crucial to take into account the changes in thickness and concentration associated with a change of velocity when performing the calculation
of the Jacobian times a vector. This operation is performed correctly in our
BDF2-IMEX-RK2 as can be seen in Fig. 6a. This figure shows that the mean
number of Newton iterations is about the same with BDF2-IMEX-RK2 than
it is with the SIT scheme (it is even a little lower). To reinforce this conclusion, we show in Fig. 8 a typical nonlinear evolution of the L2-norm of
the residual for BDF2-IMEX-RK2 and for the SIT schemes. The time step
is 30 min and the resolution is 20 km. Both schemes exhibit a very similar
nonlinear convergence. They both need 12 Newton iterations to reach the
nonlinear convergence criterion ( nl = 10 6 ).
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As in Lipscomb et al. [2], we found that the 2-D model is less sensitive
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687
688
than the 1-D model to the SIT instability. The BDF2-IMEX-RK2 scheme is
nevertheless useful as the SIT instability is more severe as the grid is refined
and when using a multi-category sea ice model [2]. Note that our method
could easily be applied to a multi-category model. Furthermore, a sea ice
model using a yield curve having less shear strength than the standard elliptical yield curve would also be more exposed to this instability and would
therefore benefit from the more stable BDF2-IMEX-RK2 scheme.
689
690
691
692
693
An obvious extension to this work would be to develop a second-order
scheme that would also include thermodynamic processes. To do so, the
predictor-corrector approach would include the source terms and would become
(h⇤
hn 1 )
=
t/2
r · (un 1 hn 1 ) + Sh (hn 1 , An 1 ),
(hn
694
1
hn 1 )
= r · (un 2 h⇤ ) + Sh (h⇤ , A⇤ ),
t
⇤
n
where A and A would be obtained in a similar way.
(31)
(32)
695
696
697
698
699
700
701
Another improvement would be to replace our di↵usive first-order in space
upstream scheme by a more sophisticated advection operator. For example,
second-order accuracy in space could also be achieved by using the remapping scheme of Lipscomb and Hunke [31]. Note that a stabilization method
(di↵erent time-stepping approach) may be required as higher order advection
schemes are less di↵usive than a first-order upstream operator.
702
703
704
705
706
707
708
709
The JFNK solver is remarkably robust in longer simulations (five years).
At 40-km resolution, JFNK did not fail for either the SIT or the BDF2IMEX-RK2 integration scheme. At 20-km resolution, convergence was not
reached on rare occasions for both integration schemes. With SIT, JFNK
had a failure rate as low as 0.027 % while JFNK with the BDF2-IMEX-RK2
scheme failed for only 0.025 % of the time levels (this is slightly smaller than
for SIT but probably not statistically significant).
710
711
712
Even though these failure rates are very small and when a failure occurs
it usually a↵ects only a few grid cells (not shown), the increase in the failure
25
713
714
715
716
rates with resolution indicates that further work is needed to improve the robustness. A more sophisticated approach than the linesearch method might
help (e.g. [32]) but we also suspect that our preconditioning approach might
need to be revisited as we refine the grid.
717
718
719
720
721
722
723
724
725
726
Indeed, as the spatial resolution increases, the rheology term makes the
problem more and more nonlinear. We have observed occasional failures of
the preconditioned FGMRES at 10-km resolution for a linear tolerance of
0.1. To improve our preconditioning operator, we are currently working on
using the MultiLevel (ML) preconditioner from the Trilinos library [33]. It is
possible, however, that this might not be sufficient and that we might have to
reconsider the use of the Picard matrix for the preconditioning step. In other
words, our preconditioning matrix might have to be closer to the Jacobian
matrix than what the Picard matrix is.
727
728
729
730
731
732
733
734
735
736
737
738
This study was done using a serial code. Losch et al. [11] have recently
implemented a parallel JFNK solver for sea ice dynamics. They have demonstrated that the scaling of JFNK with a similar line relaxation approach for
the preconditioner is almost as good as for other solvers (Picard and EVP);
in their case for domain decompositions of up to 1000 CPUs. There is no
reason to believe that our BDF2-IMEX-RK2 approach would not exhibit
similar performances as the additional thickness and concentration calculations performed in the Newton loop are explicit and do not require extra
communication overheads. Using a di↵erent preconditioner (such as ML)
might lead to an improved scalability of JFNK. This is the subject of future
work.
739
740
741
742
743
744
Appendix A: Undamped oscillation with a Crank-Nicolson approach
By centering in time (at n 12 ) the terms in the momentum equation, a
Crank-Nicolson approach also leads to second-order accuracy (not shown).
However, as explained here, it can lead to an undamped oscillation in zones
with little ice. With this approach, the u and v equations are written as
⇢h
n
1
2
(un
un 1 )
= ⇢hn
t
1
2
fv
n
1
2
n
+⌧au
1
2
n
⌧wu
1
2
+
@
n
11
1
2
n
(Pp
@x
1
2)
+
@
n
12
1
2
n
(Pp
@y
1
2)
,
(33)
26
⇢h
n
1
2
(v n
746
where hn
that the
747
and that
745
vn 1)
=
t
1
2
⇢h
n
1
2
fu
n
1
2
n
+⌧av
1
2
n
⌧wv
1
2
+
@
n
22
1
2
n
(Pp
+
@
n
12
1
2
n
(Pp
1
2)
@x
(34)
hn +hn 1
n 12
n 12
n 12
=
and A
,u
and v
are similarly defined. Note
2
1
1
and the water stress components are functions of un 2 and v n 2
ij
n
Pp
1
2
= P ⇤ hn
1
2
exp[ C(1
An
1
2
@y
1
2)
)].
748
749
750
751
752
753
754
Assuming a region with very thin ice, the balance of force is then between the water stress and the wind stress. To explain the oscillation, we
further simplify the problem by setting the water turning angle to zero and
by assuming that the ocean is at rest and that the wind is blowing from the
west (such that the ice velocity is positive). The momentum balance then
becomes
u n + un 1 2
),
(35)
2
Assume that the wind stress was zero before such that un 1 = 0 and that
after that it is constant and equal to ⌧au . The velocity at time level n is then
r
⌧au
n
u =2
,
(36)
⇢w Cdw
n
⌧au
755
756
757
759
760
761
762
763
= ⇢w Cdw (
while at n + 1 it is equal to
u
758
1
2
n+1
=2
r
⌧au
⇢w Cdw
un = 0,
(37)
and we q
find that un+2 = un , i.e., the solution oscillates between two values:
0 and 2 ⇢w⌧Caudw . This undamped oscillation is more severe when using large
time steps as a significant time di↵erence between two time levels is more
likely to lead to a large change in the wind stress. This oscillation is not
observed when using the second-order backward di↵erence time integration
approach.
764
27
,
765
766
767
768
769
770
771
772
Appendix B: Truncation error analysis
We perform a truncation error analysis similar to the one described in
Kadioglu and Knoll [14]. We assume a 1-D problem, that the velocity is positive, that the concentration is 1 everywhere and that the viscous coefficients
are constant in space and in time. The replacement closure (equation (9)) is
not used such that P = Pp . We also assume that the Newton iteration has
already converged such that uk = un and hk = hn . The momentum equation
is then given by
@u
@ 2 u 1 @P
= R = ⌧a Cu2 + ⇣ 2
,
(38)
@t
@x
2 @x
where C = ⇢w Cdw , P = P ⇤ h and R is just the sum of all the terms on the
2
RHS. To simplify the notation, we introduce Lu (u) = @@xu2 and Lp (P ) = @P
.
@x
The continuity equation for h is
⇢h
773
774
775
@h
=
@t
776
@(uh)
,
@x
for which we introduce the operator Luh (uh) =
(39)
@(uh)
.
@x
777
778
779
At time level n we solve with our BDF2-IMEX-RK2 method the following
equations
⇢hn (
3un
2
2un
hn = hn
780
with un
1
2
1
+
un 2
)=
2
tLuh (un
1
2
tRn ,
h⇤ ),
(40)
(41)
and h⇤ given by
u
n
1
2
(un + un 1 )
=
,
2
(42)
t
Luh (un 1 hn 1 ).
(43)
2
We use the following Taylor series to express un as a function of un 1
h⇤ = hn
781
1
un = un
1
+
t
1
@un
@t
1
+
28
t 2 @ 2 un
2
@t2
1
+ O( t3 ).
(44)
782
783
784
785
786
787
788
789
We now prove that our BDF2-IMEX-RK2 method leads to second-order
accuracy in time for the calculation of the velocity and the thickness. If h
and u are both second-order accurate in time, their product is also secondorder accurate in time. We can demonstrate this by starting from equation
(40) and then by using the other equations we introduced above (the LHS of
equation (40) is expressed in terms of products of h and u). Using equation
(44) and also a Taylor expansion around un 1 for un 2 , the LHS of equation
(40) can be written as
3
⇢hn (un
2
1
⇢hn (un
2
790
791
n 1
⌧✏ = ⇢ h
795
797
1
1
1
+ O( t3 ) .
(45)
(46)
tLuh (u
n
1
2
⇤
h)
i
t
@un
@t
1
+
t2
@ 2 un
@t2
1
+ O( t3 ) .
(47)
n 1
tLuh (u
n
1
2
⇤
h)
i
t
@un
@t
1
+
t2
@ 2 un
@t2
1
+ O( t3 )
tRn ,
(48)
where Rn is expanded below. The terms can be rearranged such that one
obtains
@un 1
+O( t3 )
tRn .
@t
@t
(49)
Using equations (41) and (44) and introducing a Taylor series for the
wind stress, Rn can be written as
⌧✏ =
796
t 2 @ 2 un 1
) ⇢hn 2un 1 +
2
@t2
t 2 @ 2 un 1
+
) + O( t3 ),
2
2
@t
+
From the latest equation, the truncation error (⌧✏ ) can be obtained by
subtracting the RHS of equation (40) from expression (47)
h
794
1
t
Substituting hn from equation (41) in (46) we get
⇢ h
793
@un
@t
@un
t
@t
+
which after regrouping the terms becomes

@un 1
@ 2 un
n
⇢h
t
+ t2
@t
@t2
h
792
1
t⇢hn
1 @u
n 1
+ t2 ⇢hn
[email protected]
2 n 1
u
@t2
29
t2 ⇢Luh (un
1
2
h⇤ )

@⌧an 1
@un
R
+ t
C (un 1 )2 + 2 tun 1
@t
@t
@un 1
P⇤
+ t⇣Lu (
)
Lp (hn ) + O( t2 ).
@t
2
Using again equation (41) for hn , we get
n
798
R
n
=⌧an 1
=⌧an 1
+
799
800
Simplifying and using Luh (un
(51) we get
1
+
t
@⌧an
@t
1
2
1
1 @u
2 tCun
n 1
 @tn
tP ⇤
@h
Lp
2
@t
where we have used the fact that Luh (un 1 hn 1 ) =
can write the previous equation as
n
R =R
(50)
h⇤ ) = Luh (un 1 hn 1 ) + O( t) in equation
@
t ⇣Lu (un 1 )
@t
+
802
+ ⇣Lu (un 1 )

@⌧an 1
@un 1
+ t
C (un 1 )2 + 2 tun 1
+ ⇣Lu (un 1 )
@t
@t
i
1
@un 1
P⇤
tP ⇤ h
t⇣Lu (
)
Lp (hn 1 ) +
Lp Luh (un 2 h⇤ ) + O( t2 ),
@t
2
2
(51)
Rn =Rn
801
1
n 1

@
+ t
⌧an
@t
1
1
(52)
+ O( t2 ),
@hn
@t
1
. Rearranging, we
P⇤
Lp (hn 1 ) + O( t2 ).
2
(53)
so we can write
C(un 1 )2 + ⇣Lu (un 1 )
803
The term inside the brackets is just Rn
804
@Rn 1
+ O( t2 ).
@t
We replace Rn in equation (49) using equation (54) and obtain
Rn = Rn
⌧✏ = t⇢h
n 1 @u
tR
n 1
@t
n 1
+
2
1
2
t ⇢h
@Rn
t
@t
+
n [email protected]
1
t
2 n 1
u
@t2
1
t ⇢Luh (u
3
+ O( t ).
30
2
n
1
2
@un
h)
@t
⇤
(54)
1
(55)
805
1
2
Using again Luh (un

⌧✏ = t ⇢h
n 1 @u
n 1
R
@t
h⇤ ) = Luh (un 1 hn 1 ) + O( t), we can write
n 1
+
3
+ O( t ).
t
2

⇢hn
[email protected]
2 n 1
u
@t2
+⇢
@hn 1 @un
@t
@t
1
@Rn
@t
1
(56)
806
807
Using equation (38), we can eliminate the O( t) terms. We now use
2
@h
= h @@t2u + @u
to get
@t @t
@ [email protected]
@t @t
⌧✏ =
t
2

⇢
@ n
h
@t
1 @u
n 1
@t
809
810
@un 1 @hn
@t
@t
1
+⇢
@hn 1 @un
@t
@t
1
@Rn
@t
1
+ O( t3 ).
(57)
⌧✏ =
808
⇢

@
t⇢
hn
@t
2
1 @u
n 1
@t
Rn
1
+ O( t3 ).
(58)
From equation (38) again, the first term on the right is zero and we find
that the truncation error is O( t3 ) which shows that our scheme is secondorder accurate in time.
811
812
813
814
815
Acknowledgements
We would like to thank William Lipscomb for interesting discussions
about the splitting in time instability in sea ice models. We are also grateful
to two anonymous reviewers for their helpful comments.
816
817
818
819
820
821
822
823
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34
−2
−2
10
10
(a)
−3
−3
10
10
−4
−4
10
RMSE (m)
RMSE (m)
10
−5
10
−6
−5
10
−6
10
10
x=40 km
−7
x=20 km
−7
10
10
SIT
IMEX
BDF2−IMEX−RK2
−8
10
(b)
0
10
1
2
10
10
Time step (min)
SIT
IMEX
BDF2−IMEX−RK2
−8
3
10
10
0
10
1
2
10
10
Time step (min)
3
10
Figure 4: RMSE between an approximate solution and the reference thickness as a function
of t for spatial resolutions of 40 km (a) and 20 km (b). The black curve with triangles is
the SIT method, the red curve with diamonds is the IMEX method while the blue curves
with circles is for BDF2-IMEX-RK2.
35
−1
10
−2
10
−3
RMSEv (cm/s)
10
−4
10
−5
10
−6
10
36
−7
10
0
10
1
SIT
IMEX
BDF2−IMEX−RK2
2
10
10
Time step (min)
3
10
4
10
(a)
(b)
SIT
IMEX
BDF2−IMEX−RK2
20
Total CPU time (s)
Mean nb of Newton iterations
25
15
10
3
10
5
SIT
IMEX
BDF2−IMEX−RK2
0
0
20
40
60
80
100
120
140
160
2
10
180
0
20
40
60
80
100
120
140
160
180
Time step (min)
Time step (min)
Figure 6: (a) Mean number of Newton iterations per time level as a function of t. (b)
Total CPU time as a function of t. These two quantities were calculated for the last 12
h of the integration. The black curve with triangles is the SIT method, the red curve with
diamonds is the IMEX method while the blue curves with circle is for BDF2-IMEX-RK2.
The spatial resolution is 20 km.
37
(a)
(b)
(c)
(d)
Figure 7: (a) Reference solution thickness field (in m) on 18 January 2002 00Z. This field
is capped to 4 m on the figure to see more details. (b) Di↵erence (in m) between the
approximate solution obtained with SIT with t = 90 min and the reference solution. (c)
Di↵erence (in m) between the approximate solution obtained with BDF2-IMEX-RK2 with
t = 90 min and the reference solution. (d) Di↵erence (in m) between the approximate
solution obtained with SIT with t = 10 min and the reference solution. The di↵erence
fields are capped to ±0.01 m. Note that the scale is di↵erent in (a). The spatial resolution
is 20 km.
38
0
10
BDF2−IMEX−RK2
SIT
−1
10
−2
L2−norm
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
5
10
15
Newton iteration
Figure 8: L2-norm on 18 January 2002 00Z as a function of the number of Newton iterations when using the SIT scheme (black curve with triangles) and the BDF2-IMEX-RK2
scheme (blue curve with circles). The time step is 30 min and the spatial resolution is 20
km.
39
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