1. Numerical Schemes for Advection Reaction Equation Ramaz Botchorishvili Faculty of Exact and Natural Sciences Tbilisi State University GGSWBS,Tbilisi, July 07-11, 2014 1

2. Outline 2  Equations  Operator splitting  Baricentric interpolation and derivative  Simple high order schemes  Divided differences  Stable high order scheme  Multischeme  Discretization for ODEs

3. Equation 3 v(t,x) - velocity f (u,t) – reaction smooth functions

4. Operator splitting 4

5. 5 LA -> ODE ->LA -> ODE-> … - first order accurate

6. Operator splitting 6 LA -> ODE ->LA -> ODE-> … - first order accurate LA -> ODE ->ODE->LA -> LA->ODE-> … - second order accurate

7. Interpolation 7

8. Lagrange interpolation 8

9. 9 Pros & cons

10. Barycentric interpolation 10

11. Barycentric interpolation 11

12. Barycentric interpolation 12 Advantages  Efficient in terms of arithmetic operations  Low cost for introducing or excluding new nodal points = variable accuracy

13. Baricentric derivative 13

14. Baricentric derivative 14 Advantages:  Easy for implementing  Arithmetic operations  High order accuracy

15. High order scheme for LA 15 Approximation for N=1, 2 nd order N=2, 4th order N=4, 8 th order …

16. High order scheme for LA 16 First order accurate in time, 2n order accurate in space

17. High order scheme for LA 17 First order accurate in time, 2n order accurate in space Possible Problems conservation & stability

18. Firs order upwind 18

19. Firs order upwind 19 Numerical flux functions

20. Firs order upwind 20 Numerical flux functions Properties 1.Consistency 2.Conditional stability (CFL) 3.First order in space and in time 4.conservative

21. Firs order upwind 21 Numerical flux functions Properties 1.Consistency 2.Conditional stability (CFL) 3.First order in space and in time 4.conservative

22. High order conservative discretisation 22

23. High order conservative discretisation 23 Harten, Enquist, Osher, Chakravarthy: given function in values in nodal points, how to interpolate at cell interfaces in order to get higher then 2 accuracy

24. Special interpolation/reconstruction procedure 24

25. Special interpolation/reconstruction procedure 25

26. Special interpolation/reconstruction procedure 26

27. Special interpolation/reconstruction procedure 27

28. Special interpolation/reconstruction procedure 28

29. Special interpolation/reconstruction procedure 29

30. Special interpolation/reconstruction procedure 30

31. Special interpolation/reconstruction procedure 31

32. Special interpolation/reconstruction procedure 32

33. Special interpolation/reconstruction procedure 33

34. Special interpolation/reconstruction procedure 34

35. Special interpolation/reconstruction procedure 35

36. Special interpolation/reconstruction procedure 36

37. Special interpolation/reconstruction procedure 37 High order accurate approximation

38. Special interpolation/reconstruction procedure 38 High order accurate approximation

39. Components of high order scheme 39 Discretization of the divergence operator baricentric interpolation baricentric derivative ENO type reconstruction procedure (Harten, Enquist, Osher, Chakravarthy): Given fluxes in nodal points Interpolate fluxes at cell interfaces in such a way that central finite difference formula provides high order (higher then 2) approximation Use adaptive stencils to avoid oscillations

40. Adaptation of interpolation 40 Use adaptive stencils to avoid oscillations interpolate with high order polynomial in all cell interfaces inside of the stencil of the polynomial If local maximum principle is satisfied then value at this cell interface is found If local maximum principle is not satisfied then change stencil and repeat interpolation procedure for such cell interfaces If after above procedure local maximum principle is not respected then reduce order of polynomial and repeat procedure for those interfaces only

41. Convergence in one space dimension 41  Algorithm ensures  Uniform bound of approximate solutions  Uniform bound of total variation  Conclusions Approximate solution converge a.e. to solution of the original problem

42. Extension to higher spatial dimension 42  Cartesian meshes:  strightforward  Hexagonal meshes:  Directional derivates => div needs three directional derivatives in 2D  Implementation with baricentric derivatives without adaptation procedure  See poster ( Tako & Natalia)  Implementation with adaptation – not yet

43. Better ODE solvers 43  Different then polinomial basis fanctions, e.g approach B.Paternoster, R.D’Ambrosio

44. 44 Thank you