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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

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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.

.

Then the local truncation of CNLF satisfies

Denote

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

1st order

(i) The common splitting condition is not necessary for 2nd order CNLF

convergence. What is the right condition?

(ii) But why only 1st order when

IMEX-Euler is used to start up?

2nd order

-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,
- IMEX-Euler is 1st-order convergent
- CNLF with IMEX-Euler startis 1st-order convergent
- 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 thegeneral ansatz

The splitting condition

- The common splitting condition requires

- However

Hence

fails

- The new splitting condition

is to be interpreted as a discrete spatial integration

which “removes” the factor

Stability

- Stability analysis of IMEX methods normally requires

commuting operators. However,

which isnot 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 sizeτ = h with

hthe coarse grid size

1st order

--- : 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 !!!

1/h

Illustration I (uniform grid, random S)

Uniform grid and S randomly chosen as

Global errors at t = 0.25

Step sizeτ = 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 = 1)

U(x,y,t = 0)

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