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Numerical Propagator Method Computations for Parabolic Equations

Numerical Propagator Method Computations for Parabolic Equations. Janis Rimshans Institute of Mathematics and Computer Science, University of Latvia, Riga, LV-1459, Latvia. Workshop on Time Integration of Evolution Equations September, 200 7 , Innsbruck, Austria. Contents.

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Numerical Propagator Method Computations for Parabolic Equations

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  1. Numerical Propagator Method Computations for Parabolic Equations Janis Rimshans Institute of Mathematics and Computer Science, University of Latvia, Riga, LV-1459, Latvia Workshop on Time Integration of Evolution Equations September, 2007, Innsbruck, Austria

  2. Contents Problem Formulation Numerical Propagator Method Cauchy problem Stability analysis Numerical results Analytical method Conclusions

  3. Problem Formulation , ; , , Initial condition: ; , Newton boundary conditions: , , , ; ,

  4. Problem Formulation , , , , ; Boundary conditions coefficients: ; !! ; , , Solvability conditions: , , , , ,

  5. Numerical Propagator Method Equation: , , . , Propagator methodsubstitution: , , , Equation for solution:

  6. Numerical Propagator Method Semi-implicit propagator central difference scheme: , Seven point central difference operator . , ,

  7. Cauchy Problem for ADR Equation ; Semi-implicit propagator central difference scheme: Scheme is unconditionally monotone since the coefficients of the scheme satisfy the maximal principle conditions !!

  8. Stability Analysis for Semi-implicit Central Difference Scheme , ,

  9. Stability Analysis for Semi-implicit Central Difference Scheme von Neumann approachgives: If for which then stability restriction is:

  10. Stability Analysis for Semi-implicit Central Difference Scheme limiting expressionis: , ,

  11. Stability Analysis for Semi-implicit Propagator Difference Scheme where and are solution and forcing term estimations,respectively. If , then: , where !! .

  12. Stability Analysis for Semi-implicit Propagator Difference Scheme , is Domenico where If analytical solution, then:

  13. Stability Analysis for Semi-implicit Propagator Difference Scheme von Neumann approachgives: Propagator semi-implicit scheme is absolutely stable, if !! For other cases: or:

  14. Cauchy Problem: Numerical Results Limiting time step k Re Propagator difference scheme Central difference scheme 101000 0.1 s10-10 s 10100 0.2 s10-7 s 1010 0.4 s10-4 s 10 1 1 s 0.2 s 100.1 0.9 s

  15. Analytical Method By applying substitution: , , , and using Green function approach analytical solution consistent with solubility conditions is obtained: J.Rimshans and Sh.Guseynov,Numerical Propagator Method Solutions for the Linear Parabolic Initial-Boundary Value Problems, In Proc. of the 6th InternationalCongress on Industrial and Applied Mathematics - ICIAM 2007, Zurich, Switzerland,2007, IC/CT3029/025

  16. Analytical Method

  17. Conclusions 1. A new propagator finite volume difference scheme is proposed. 2. It is shown, that stability restrictions for the propagator scheme become more weaker in comparison to traditional semi-implicit central difference schemes. There are some regions of ADR coefficients, for which elaborated propagator difference scheme becomes absolutely stable. 3. It is proven that the scheme is unconditionally monotonic. 4. The scheme has the first order in time and the second order truncation errors in space. 5. The scheme can be easy extended to the solution of multidimensional non-steady problems. 6. Using Green function approach analytical solution is obtained for 3D problem with Newton boundary conditions. 7. In addition, in order to fulfil, that , such limitations for velocities is found: , .

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