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IMEX Methods for Advection-Diffusion-Reaction Equations

IMEX Methods for Advection-Diffusion-Reaction Equations. Speaker : Volha Shchetnikava Adviser: dr.ir.J.H.M. ten Thije Boonkkamp. Eindhoven 2008. Contents. 1. Introduction A-D-R equations 2. Implicit -explicit (IMEX) methods Description

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IMEX Methods for Advection-Diffusion-Reaction Equations

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  1. IMEX Methods for Advection-Diffusion-Reaction Equations Speaker : Volha Shchetnikava Adviser: dr.ir.J.H.M. ten Thije Boonkkamp Eindhoven 2008

  2. Contents 1. Introduction • A-D-R equations 2. Implicit-explicit (IMEX) methods • Description • Stability of IMEX methods • Why IMEX? 3. IMEX linear multistepmethods • Design of IMEX linear multistep methods • Examples 4. Numerical Experiments 5. Conclusions

  3. 1. IntroductionAdvection-Diffusion-ReactionEquations Model problem where u(x,t) - concentration of a certain species, a(x,t) - velocity of flowing medium, d(x,t) - diffusion coefficient f(u) - source, sink function

  4. 1. IntroductionAdvection-Diffusion-ReactionEquations Fields of application • Environmental modeling (weather forecast, water flow) • Mathematical biology (bacterial growth, tumor growth) • Chemistry • Mechanics

  5. 1. IntroductionAdvection-Diffusion-ReactionEquations UsingnumericaltechniqueisThe MethodofLines (MOL). MOL algorithm: • Discretize all spatialoperators • Obtain a systemof ODEs • Integrate ODEs system in time Advantages of MOL: • Spatialdiscretizationand time integrationaretreatedseparately • Spatialdiscretization - easy tocombine different schemes • Time integration - freetochoosesuitablemethod

  6. 2. Implicit-Explicit Methods Description IMEX method - different integrators to different terms. System of ODEs

  7. 2. Implicit-Explicit Methods Description

  8. 2. Implicit-Explicit Methods Description

  9. 2. Implicit-Explicit MethodsStabilityof IMEX method

  10. 2. Implicit-explicit (IMEX) methodsStability of IMEX methods

  11. 2. Implicit-explicit (IMEX) methodsStability of IMEX methods

  12. 2. Implicit-explicit (IMEX) methodsWhy IMEX? Why not fully explicit method? • Stability will require very small step sizes for stiff sources Why not fully implicit method? • For advection descretizations the implicit relations are hard to solve • High computational cost Why IMEX? • IMEX show a significant computational savings due to less restricted time step size • The method remains stable for time steps much larger than those that would be possible for a purely explicit method. • Very effective in many situations • Easy to apply

  13. 3. IMEX linear multistep methodsDesign of IMEX linear multistep methods

  14. 3. IMEX linear multistepmethodsExamples .

  15. 3. IMEX linear multistepmethodsExamples

  16. 3. IMEX linear multistepmethodsExamples

  17. 4. Numericalexperiments

  18. 4. Numericalexperiments f0_unst2.avi f1_unst1.avi f0_st1.avi f1_st2.avi adams_5_2.avi f1_25_2.avi adams_25_3.avi

  19. 5. Conclusions • Significant computational saving • Stable for time steps larger then for explicit method • IMEX schemes are not universal for all problems • Very effective in many situations • IMEX BDF is more stable then IMEX -CNLF

  20. Thank you!

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