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

2. Outline TU/e Eindhoven, 27th June, 2002 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

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

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

5. TU/e Eindhoven, 27th June, 2002 Time Integration in the Yee-method So-called leapfrog scheme is used: E 0 H 1/2 E 1 H 3/2 E 2 H 5/2 E 3 H 7/2... Stability condition:

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

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

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

9. TU/e Eindhoven, 27th June, 2002 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

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

11. TU/e Eindhoven, 27th June, 2002 2D example for splitting  12 34 76 5 8 H z x E y H 9

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

13. TU/e Eindhoven, 27th June, 2002 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.

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

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

16. TU/e Eindhoven, 27th June, 2002 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 if is fixed ( ), then the relation is true for all k with a constant K independent of k.

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

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

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

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

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

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

23. TU/e Eindhoven, 27th June, 2002 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).

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