imex methods for advection-diffusion-reaction equations
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
0,1Ω ,Ω)0,(
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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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termadvection from emanates
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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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any for stable is scheme IMEX the:
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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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scheme CNLF-IMEX
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),(2
Frog)-(Leap rulemidpoint Explicit
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3. I3. IMEX linear multistep methodsMEX linear multistep methods ExamplesExamples
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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,(
T][0,2/))sin(1(),1(
T][0,0),0(
T][0,)(0,1 ),()1(
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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
10,10,01.0,1 a
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!