jan verwer
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Convergence and Component Splitting for the Crank-Nicolson Leap-Frog Scheme. Jan Verwer. Hairer-60 Conference, Geneva, June 2009. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. Crank-Nicolson Leap-Frog (CNLF). non-stiff. stiff. - PowerPoint PPT PresentationTRANSCRIPT
Jan Verwer
Convergence and Component Splitting for the Crank-Nicolson Leap-Frog Scheme
Hairer-60 Conference, Geneva, June 2009
Crank-Nicolson Leap-Frog (CNLF)
non-stiff stiff
given
Usually IMEX-Euler for
CNLF:
Contents of this talk
- A splitting (convergence) condition justifying a wider class of splittings than normally seen in CFD
- As an example, component splitting for 1st order Maxwell-type wave equations
- Two numerical illustrations of component splitting
CNLF is a two-step IMEX scheme. Used for PDEs inCFD (method of lines). Non-stiff term then represents convection and the stiff term diffusion + reactions.
This talk is about an alternative use of CNLF:
Consistency of CNLF
We always think of semi-discrete systems
are always supposed to be derived and valid for
but suppress for convenience the spatial mesh size
Further, order terms like
Consistency of CNLF
Just for convenience we neglect spatial errors.
.
Then the local truncation of CNLF satisfies
ifIn CFD applications this splitting (convergence) condition is mostly satisfied!
Denote
Consistency of CNLF For the IMEX-Euler scheme
the splitting (convergence) condition features in the same way. That is, if
then
uniformly in the spatial mesh size
Convergence of CNLF Hence, if
and assuming stability, CNLF with Euler start will converge with order two uniformly in the spatial mesh width!
Q: is this common splitting (convergence) condition also necessary for 2nd – order convergence?
(i) The common splitting condition is not necessary for 2nd order CNLFconvergence. What is the right condition?
(ii) But why only 1st order when IMEX-Euler is used to start up?
Numerical counter example Semi-discrete 1st-order wave equation, with a splitting such that is violated (splitting details later).
-o- : Exact (or CN) start -*- : IMEX-Euler start
1st order
2nd order
We let
A new splitting (convergence) condition
First the linear case:
Proofs rest on local error cancellation of terms that cause order reduction if is violated. The cancellation fails at the first CNLF step when IMEX-Euler is used to compute .
(n)
Thm. Assume stability and condition (n). Then, uniformly in h,(i) IMEX-Euler is 1st-order convergent (ii) CNLF with IMEX-Euler start is 1st-order convergent(iii) CNLF with “exact start” is 2nd-order convergent
A new splitting (convergence) condition
The non-linear case:
The new condition reads
Component splittingDiscussed for linear, semi-discrete 1st order wave equations
CNLF:
where
with S a diagonal matrix satisfying the general ansatz
The splitting condition
- However
- The common splitting condition requires
- The new splitting condition
is to be interpreted as a discrete spatial integration which “removes” the factor
Hencefails
Stability
- All we can say is that
- Stability analysis of IMEX methods normally requires commuting operators. However,
which is not true!
which regarding stability is necessary for the LF part and sufficient for the CN part in CNLF
- Experience: runs are stable for the maximal stable step size for the LF part
Numerical illustration I
The component splitting matrix S is chosen in the form
Illustration I (piecewise uniform grid)
Splitting matrix S such that LF is appliedat the coarse grid and CN at the fine grid.Factor 10 between coarse & fine grid!
Illustration I (the splitting conditions)
Plots for time t = 0
1/h
Illustration I (global errors)
--- : 2nd - order -o- : CNLF with CN start -*- : CNLF with IMEX-Euler start-+- : CN
Maximal step size τ = h with h the coarse grid size
Global errors at t = 0.25
1/hCNLF with CN startgives 2nd order
The IMEX-Euler start causes order reduction !!!
1st order
Illustration I (uniform grid, random S)
--- : 2nd order-o- : CNLF with CN start -*- : CNLF with IMEX-Euler start-+- : CN
Step size τ = h
Uniform grid and S randomly chosen as
Results are in line withthose on the non-uniform grid
Global errors at t = 0.25
Illustration II
2D Maxwelltype problemon unit square
U(x,y,t = 0) U(x,y,t = 1)
Illustration II
Strongly peaked 0.95 < d(x,y) ≤ 100. Through component splitting, we use CN near the peak (d ≥ 1) and LF else-where, to avoid the step size limitation for LF near the peak
A uniform staggered grid and 2nd order differencing with grid size h requires for LF
The following results at t = 1 are obtained with CNLF for
using only a very small amount of implicitly treated points
Illustration II
CNLF is as accurate as CN
Illustration II
nnz: number of nonzeros in linear system matrix (sparsity indicator)
Conclusions
-- Component splitting tests confirm the new CNLF convergence condition
-- Component splitting can be set up in the same way for 3D Maxwell
-- But, how practical this is for real applications, I don’t know yet