IMEX Methods forAdvection-Diffusion-Reaction Equations

Eindhoven2008

Speaker : Volha ShchetnikavaAdviser: dr.ir.J.H.M. ten Thije Boonkkamp

ContentsContents

1. Introduction A-D-R equations

2. Implicit-explicit (IMEX) methods Description Stability of IMEX methods Why IMEX?

3. IMEX linear multistep methodsDesign of IMEX linear multistep methodsExamples

4. Numerical Experiments

5. Conclusions

1. Introduction1. Introduction Advection-Diffusion-Reaction Equations 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

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1. Introduction1. Introduction Advection-Diffusion-Reaction Equations Advection-Diffusion-Reaction Equations

Fields of application

Environmental modeling (weather forecast, water flow)

Mathematical biology (bacterial growth, tumor growth)

Chemistry

Mechanics

1. Introduction1. Introduction Advection-Diffusion-Reaction Equations 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

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2. Implicit-Explicit Methods2. Implicit-Explicit Methods Description Description

IMEX method - different integrators to different terms.

System of ODEs

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2. Implicit-Explicit Methods2. Implicit-Explicit Methods Description Description

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Method -IMEX

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2. Implicit-Explicit Methods2. Implicit-Explicit Methods Description Description

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2. Implicit-Explicit Methods2. Implicit-Explicit Methods Stability of IMEX method Stability of IMEX method

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

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

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2. 2. Implicit-explicit (IMEX) methods Implicit-explicit (IMEX) methods Why IMEX?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

3. I3. IMEX linear multistep methodsMEX linear multistep methods Design of IMEX linear multistep methodsDesign of IMEX linear multistep methods

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3. I3. IMEX linear multistep methodsMEX linear multistep methods ExamplesExamples

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4. 4. Numerical experimentsNumerical experiments

1,02/1)0,(

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T][0,0),0(

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4. Numerical experiments

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10,10,10,5 a

10,10,10,25 a

5. Conclusions5. 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

Thank you!