PDE stability analysis

PDE stability analysis
PDE Solver Stability and
Multigrid Methods
Review of Finite Differences Method
Finite Differences vs. Method of Lines
• Finite difference method yields recurrence relation:
• Compare to method of lines with spatial mesh size ∆x:
• Finite difference method is equivalent to solving each yi
using Euler’s method with h= ∆t
Stability of Finite Differences
• Rewrite as:
• For forward Euler to be stable, must have ∆t <= (∆x)2 / 2c
• Quite restrictive on ∆t!
– Equivalent approach for advection equation turns out to be
unconditionally unstable
Alternative Stencils
• Unconditionally stable with respect to ∆t
• (Again, no comment on accuracy)
Lax Equivalence Theorem
• For a well-posed linear PDE, two necessary and
sufficient conditions for finite difference scheme to
converge to true solution as ∆x and ∆t → 0 :
– Consistency: local truncation error goes to zero
– Stability: solution remains bounded
– Both are required
• Consistency derived from soundness of
approximation to derivatives as ∆t → 0
– i.e., does numerical method approximate the correct PDE?
• Stability: exact analysis often difficult (but less difficult
than showing convergence directly)
Reasoning about PDE Stability
• Matrix method
– Shown on previous slides
• Domains of dependence
Domains of Dependence
• Courant–Friedrichs–Lewy (CFL) condition: For each mesh point,
the domain of dependence of the PDE must lie within
the domain of dependence of the finite difference scheme
Dependence of PDE
Dependence
of scheme
Notes on CFL Conditions
• Encapsulated in “CFL Number” or “Courant
number” that relates ∆t to ∆x for a particular
equation
• CFL conditions are necessary but not sufficient
• Can be very restrictive on choice of ∆t
• Implicit methods may not require low CFL
number for stability, but still may require low
number for accuracy
Multigrid Methods
See Heath slides
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