Convergence and Component Splitting
for the Crank-Nicolson Leap-Frog Scheme
Jan Verwer
Hairer-60 Conference, Geneva, June 2009
Crank-Nicolson Leap-Frog (CNLF)
non-stiff
stiff
CNLF:
given
Usually IMEX-Euler for
Contents of this talk
CNLF is a two-step IMEX scheme. Used for PDEs in
CFD (method of lines). Non-stiff term then represents
convection and the stiff term diffusion + reactions.
This talk is about an alternative use of CNLF:
- 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
Consistency of CNLF
We always think of semi-discrete systems
but suppress for convenience the spatial mesh size
Further, order terms like
are always supposed to be derived and valid for
Consistency of CNLF
Just for convenience we neglect spatial errors.
Denote
.
Then the local truncation of CNLF satisfies
if
In CFD applications this
splitting (convergence)
condition is mostly satisfied!
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?
Numerical counter example
Semi-discrete 1st-order wave equation, with a splitting such that
is violated (splitting details later).
We let
(i) The common splitting condition
is not necessary for 2nd order CNLF
convergence. What is the right
condition?
1st order
2nd order
(ii) But why only 1st order when
IMEX-Euler is used to start up?
-o- : Exact (or CN) start
-*- : IMEX-Euler start
A new splitting (convergence) condition
First the linear case:
(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
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
.
A new splitting (convergence) condition
The non-linear case:
The new condition reads
Component splitting
Discussed for linear, semi-discrete 1st order wave equations
CNLF:
where
with S a diagonal matrix satisfying the general ansatz
The splitting condition
- The common splitting condition requires
- However
- The new splitting condition
is to be interpreted as a discrete spatial integration
which “removes” the
factor
Hence
fails
Stability
- Stability analysis of IMEX methods normally requires
commuting operators. However,
which is not true!
- All we can say is that
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 applied
at 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)
Global errors at t = 0.25
Maximal step size τ = h with
h the coarse grid size
1st order
1/h
--- : 2nd - order
-o- : CNLF with CN start
-*- : CNLF with IMEX-Euler start
-+- : CN
CNLF with CN start
gives 2nd order
The IMEX-Euler start
causes order reduction !!!
Illustration I (uniform grid, random S)
Uniform grid and S randomly chosen as
Global errors at t = 0.25
Step size τ = h
--- : 2nd order
-o- : CNLF with CN start
-*- : CNLF with IMEX-Euler start
-+- : CN
Results are in line with
those on the non-uniform grid
Illustration II
2D Maxwell
type problem
on 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 elsewhere, 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