Jan Verwer Convergence and Component Splitting for the Crank-Nicolson Leap-Frog Scheme Hairer-60 Conference, Geneva, June 2009 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A

Crank-Nicolson Leap-Frog (CNLF) non-stiffstiff 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 1 st order Maxwell-type wave equations - Two numerical illustrations of component splitting 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:

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 if In 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 2 nd – order convergence?

(i) The common splitting condition is not necessary for 2 nd order CNLF convergence. What is the right condition? (ii) But why only 1 st order when IMEX-Euler is used to start up? Numerical counter example Semi-discrete 1 st -order wave equation, with a splitting such that is violated (splitting details later). -o- : Exact (or CN) start -*- : IMEX-Euler start 1 st order 2 nd 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 1 st -order convergent (iii)CNLF with “exact start” is 2 nd -order convergent

A new splitting (convergence) condition The non-linear case: The new condition reads

Component splitting Discussed for linear, semi-discrete 1 st 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 Hence fails

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 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) --- : 2 nd - 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 = /h CNLF with CN start gives 2 nd order The IMEX-Euler start causes order reduction !!! 1 st order

Illustration I (uniform grid, random S) --- : 2 nd 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 with those on the non-uniform grid Global errors at t = 0.25

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 else- where, to avoid the step size limitation for LF near the peak A uniform staggered grid and 2 nd 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