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Numerical Schemes for Advection Reaction Equation

Numerical Schemes for Advection Reaction Equation. Ramaz Botchorishvili Faculty of Exact and Natural Sciences Tbilisi State University GGSWBS , Tbilisi , July 07 -11, 2014. Outline. Equations Operator splitting Baricentric interpolation and derivative Simple high order schemes

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Numerical Schemes for Advection Reaction Equation

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

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

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

  4. Operator splitting

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

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

  7. Interpolation

  8. Lagrange interpolation

  9. Lagrange interpolation Pros & cons

  10. Barycentric interpolation

  11. Barycentric interpolation

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

  13. Baricentricderivative

  14. Baricentricderivative • Advantages: • Easy for implementing • Arithmetic operations • High order accuracy

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

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

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

  18. Firs order upwind

  19. Firs order upwind Numerical flux functions

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

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

  22. High order conservative discretisation

  23. High order conservative discretisation 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

  25. Special interpolation/reconstruction procedure

  26. Special interpolation/reconstruction procedure

  27. Special interpolation/reconstruction procedure

  28. Special interpolation/reconstruction procedure

  29. Special interpolation/reconstruction procedure

  30. Special interpolation/reconstruction procedure

  31. Special interpolation/reconstruction procedure

  32. Special interpolation/reconstruction procedure

  33. Special interpolation/reconstruction procedure

  34. Special interpolation/reconstruction procedure

  35. Special interpolation/reconstruction procedure

  36. Special interpolation/reconstruction procedure

  37. Special interpolation/reconstruction procedure High order accurate approximation

  38. Special interpolation/reconstruction procedure High order accurate approximation

  39. Components of high order scheme • 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 • 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 • 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 • 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 • Different then polinomial basis fanctions, e.g approach B.Paternoster, R.D’Ambrosio

  44. Thank you

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