Uniform Treatment of Numerical Time-Integrations of the Maxwell Equations R. Horváth

1. Uniform Treatment of Numerical Time-Integrations of the Maxwell Equations R. Horváth TU/e, Philips Research Eindhoven Scientific Computing in Electrical Engineering Eindhoven, The Netherlands June, 23rd - 28th, 2002 Eindhoven, 27th June, 2002 TU/e

2. Outline • Introduction, Yee-method, problems • Operator splitting methods • Splitting of the semi-discretized Maxwell equations (Yee, NZCZ, KFR) • Proof of the unconditional stability of the NZCZ-method • Comparison of the methods Eindhoven, 27th June, 2002 TU/e

3. Maxwell Equations Eindhoven, 27th June, 2002 TU/e

4. Space Discretization in the Yee-method Second order spatial discretization using staggered grids. Eindhoven, 27th June, 2002 TU/e

5. E0 H1/2E1 H3/2 E2 H5/2 E3 H7/2... Time Integration in the Yee-method So-called leapfrog scheme is used: Stability condition: Eindhoven, 27th June, 2002 TU/e

6. Operator Splitting Methods I. Sequential splitting (S-splitting): Eindhoven, 27th June, 2002 TU/e

7. Operator Splitting Methods II. Splitting error: s: number of the sub-systems Lemma: For the S-splitting method (1st order): Eindhoven, 27th June, 2002 TU/e

8. Operator Splitting Methods III. S-splitting: Other splittings: Strang-splitting (second order): Fourth order splitting: Eindhoven, 27th June, 2002 TU/e

9. Semi-discretization of the Maxwell Equations Applying the staggered grid spatial discretization, the Maxwell equations can be written in the form: :skew-symmetric, sparse matrix, at most four elements per row. The elements have the form . : consists of the field components in the form Eindhoven, 27th June, 2002 TU/e

10. Splitting I. (Yee-scheme, 1966) We zero the rows of the electric field variables We zero the rows of the magnetic field variables Lemma: Eindhoven, 27th June, 2002 TU/e

11. 7 6 1 9 2 8 E 5 z 3 4 H y H D x 2D example for splitting Eindhoven, 27th June, 2002 TU/e

12. 2D example - Yee-method Eindhoven, 27th June, 2002 TU/e

13. Splitting II. (KFR-scheme, 2001) The matrices are skew-symmetric matrices, where the exponentials exp(A.K) can be computed easily using the identity: Stability: the KFR-method is unconditionally stable by construction. Eindhoven, 27th June, 2002 TU/e

14. 2D example - KFR-method Eindhoven, 27th June, 2002 TU/e

15. Splitting III. (NZCZ-scheme, 2000) skew-symmetric Discretization of second terms in the curl operator Discretization of first terms in the curl operator The sub-systems with and cannot be solved exactly. Numerical time integrations are needed. Eindhoven, 27th June, 2002 TU/e

16. Splitting III. (NZCZ-scheme, 2000) Solving the systems by the explicit, implicit, explicit and implicit Euler-methods, respectively, we obtain the iteration Theorem: the NZCZ-method is unconditionally stable. Proof: We show that ifis fixed ( ), then the relation is true for all k with a constant K independent of k. Eindhoven, 27th June, 2002 TU/e

17. Splitting III. (NZCZ-scheme, 2000) Eindhoven, 27th June, 2002 TU/e

18. 2D example - NZCZ-method Eindhoven, 27th June, 2002 TU/e

19. 2D example - Real Schur Decomposition No splitting error! Eindhoven, 27th June, 2002 TU/e

20. 2D examples - Comparison Eindhoven, 27th June, 2002 TU/e

21. Comparison of the methods Eindhoven, 27th June, 2002 TU/e

22. Comparison of the methods Eindhoven, 27th June, 2002 TU/e

23. Comparison of the methods What is the reason of the difference between the Yee and the KFR-method? Compare Yee and KFR1: Yee: sufficiently accurate, strict stability condition KFR: unconditionally stable, but an accurate solution requires small time-step. In the long run it is slower than the Yee-method. NZCZ: unconditionally stable. With a suitable choice of the time-step the method is faster than the others (with acceptable error). Eindhoven, 27th June, 2002 TU/e

24. Thank You for the Attention Eindhoven, 27th June, 2002 TU/e