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

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

3. 1. Introduction Advection-Diffusion-Reaction Equations 1. Introduction Advection-Diffusion-Reaction Equations 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. Introduction Advection-Diffusion-Reaction Equations 1. Introduction Advection-Diffusion-Reaction Equations Fields of application Environmental modeling (weather forecast, water flow) Mathematical biology (bacterial growth, tumor growth) Chemistry Mechanics

5. 1. Introduction Advection-Diffusion-Reaction Equations Using numerical technique is The Method of Lines (MOL). MOL algorithm : 1. Discretize all spatial operators 2. Obtain a system of ODEs 3. Integrate ODEs system in time Advantages of MOL : 1. Spatial discretization and time integration are treated separately 2. Spatial discretization - easy to combine different schemes 3. Time integration - free to choose suitable method

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

7. 2. Implicit-Explicit Methods Description 2. Implicit-Explicit Methods Description

8. 2. Implicit-Explicit Methods Description

9. 2. Implicit-Explicit Methods Stability of IMEX method

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

12. 2. Implicit-explicit (IMEX) methods Why IMEX? 2. Implicit-explicit (IMEX) methods Why 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 methods Design of IMEX linear multistep methods

14. 3. IMEX linear multistep methods Examples 3. IMEX linear multistep methods Examples.

17. 4. Numerical experiments 4. Numerical experiments

18. 4. Numerical experiments 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!