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

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Presentation transcript:

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

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

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

Operator splitting 4

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

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

Interpolation 7

Lagrange interpolation 8

9 Pros & cons

Barycentric interpolation 10

Barycentric interpolation 11

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

Baricentric derivative 13

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

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

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

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

Firs order upwind 18

Firs order upwind 19 Numerical flux functions

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

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

High order conservative discretisation 22

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

Special interpolation/reconstruction procedure 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 High order accurate approximation

Special interpolation/reconstruction procedure 38 High order accurate approximation

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

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

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

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

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

44 Thank you