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 splitting

Discussed 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