1. The Interplay of Modelling Analysis, Discretization and Numerics in Numerical Simulation Christoph Zenger Institut für Informatik TU München

2. Numerical Simulation Christoph Zenger, Informatik V, TU München 2 Numerical Simulation? Physics Mathematics Computer Science

3. Numerical Simulation Christoph Zenger, Informatik V, TU München 3 Physics: Model, Equations Mathematics: Analysis, Discretization Informatics: Program Numerical Simulation?

4. Numerical Simulation Christoph Zenger, Informatik V, TU München 4 Backward Analysis Problem: Make a round table plate using wood with radius 1m, thickness 2 cm and specific weight1g/cm³. Compute the weight of the plate Result: G = 62.832 kg Error < 1g Backward Analysis: G is the exact weight of a plate with another (slightly different) radius difference between r and < 0.1mm

5. Numerical Simulation Christoph Zenger, Informatik V, TU München 5 Data →Inexact Computation →Result ← ERROR → Backward Forward

6. Numerical Simulation Christoph Zenger, Informatik V, TU München 6 Intersection point of two straight lines Intersection point is defined only approximately (green area)

7. Numerical Simulation Christoph Zenger, Informatik V, TU München 7 Forward rounding error analysis with error norms → red area. Backward rounding error analysis (Wilkinson): solution exact for slightly modified data (straight lines) → solution in green area. Gauss-Elimination result is indeed in green area. Cramer‘s rule: result in red area!

8. Numerical Simulation Christoph Zenger, Informatik V, TU München 8 Example: Heat conduction on a thin plate 1 1 -4 1 T = 0 1 The computed solution is not exact (Discretization error). But we can apply backward analysis: Solution exact to a slightly modified heat conduction problem Physical problem Discretization

9. Numerical Simulation Christoph Zenger, Informatik V, TU München 9 old problem slightly modified problem The modified problem is again a „physical problem“! All physical laws are valid! E. g.: The maximal temperature is always on the boundary!

10. Numerical Simulation Christoph Zenger, Informatik V, TU München 10 This is not true in general for a Finite Element solution (e.g. bilinear rectangular f.e). The finite element solution does not correspond to a slightly modified heat conduction problem! Physical laws are invalid. E.g.: The maximal temperature is not always on the boundary! Consequence may be disastrous: E. g. Boundary 100°C for water in a pot.

11. Numerical Simulation Christoph Zenger, Informatik V, TU München 11 Example 2: clamped membrane membrane membrane enforced with sticks We can show: The FE-Method with bilinear elements yields the exact solution to the modified problem. Here the backward analysis is possible for the FE –method! Joints (same equations as before)

12. Numerical Simulation Christoph Zenger, Informatik V, TU München 12 Example 3: Incompressible fluid flows: Navier Stokes Equations: Conservation of momentum conservation of mass Die meisten Lösungsverfahren verletzen Energie und/oder Impuls- Erhaltung!, d. h. das Verfahren löst kein physikalisch mögliches System! Mögliche Konsequenz: Die kinetische Energie steigt auch ohne Energiezufuhr. Besonders häufig bei Strömungen mit hoher Reynoldszahl, weil hier sehr wenig Energie durch Reibung verloren geht.

13. Numerical Simulation Christoph Zenger, Informatik V, TU München 13 Example 3: Incompressible fluid flows: Navier Stokes Equations: Conservation of momentum conservation of mass What about backward analysis in this case?

14. Numerical Simulation Christoph Zenger, Informatik V, TU München 14 It is not important that the modified problem is physically realizable. That was not even possible for the clamped membrane. But: The physical laws should be satisfied: In our case most important: Conservation of energy, momentum and mass! Most methods used in practice do not satisfy these conservation laws (e. g. most finite element Methods) and most textbooks don’t discuss this issue. We give an example of an acceptable method:

15. Numerical Simulation Christoph Zenger, Informatik V, TU München 15 uuuu uuu uuuu u vvv vvv vvv vvv „staggered grid“

16. Numerical Simulation Christoph Zenger, Informatik V, TU München 16 uuuu uuu uuuu u vvv vvv vvv vvv Interpolation of u (v rotated by 90°)

17. Numerical Simulation Christoph Zenger, Informatik V, TU München 17 This discrete solution satisfies the continuity equation pointwise (not only in a weak sense). As a consequence solution (if integrated exactly in time) satisfies the conservation laws. If the energy flow through the boundary is controlled, the discrete solution exists for an arbitrarily long time interval! (The solution cannot grow in an unbounded way because of bounded energy). Integration in time should be accomplished by a symplectic method.

18. Numerical Simulation Christoph Zenger, Informatik V, TU München 18 Remark: Convection has no influence on energy balance, Diffusion transforms kinetic energy to thermal energy (kinetic energy gets smaller). These properties have to be satisfied by the discrete scheme (exactly not only O(h²)!). This is especially important, if diffusion is very small (high Reynold‘s number) e.g. for turbulent flows. Extreme example in Astrophysics are accretion discs. In this case one cannot use the Eulerian approach (because interpolation is “diffusive”). Lagrange‘s method or particle methods are appropriate.

19. Numerical Simulation Christoph Zenger, Informatik V, TU München 19 Conclusion: The interplay of modeling and numerical solution is more complex than usually acknowledged. A similar situation holds for the interplay between mathematical analysis and programming or physics and programming: Hierarchical data structures ↔ Multilevel Methods ↔hierarchical visualization Parallel Numerics ↔ Iteration Strategies Discussion in the following lectures.

20. Numerical Simulation Christoph Zenger, Informatik V, TU München 20 actual state of the art in science:

21. Numerical Simulation Christoph Zenger, Informatik V, TU München 21 Diss. Diss

22. Numerical Simulation Christoph Zenger, Informatik V, TU München 22 Group work

23. Numerical Simulation Christoph Zenger, Informatik V, TU München 23 Thank you