Runge-kutta Method For.ppt 1wc6y
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Jie, Liu CASA Seminar September 17 2oo8
I.
Introduction
II.
Runge-Kutta Methods for ODE Systems
III.
Stability Analysis for the AdvectionDiffusion-Reaction Equation
IV.
Numerical Results
V.
Conclusions Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
2
The parameter a is advection velocity
is diffusion coefficient is source term coefficient Reaction term is logistic growth
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
3
Arise in many chemical and biological settings.
In hydrology , equations of this type model the transport and fate of adsorbing contaminants and microbe-nutrient systems in groundwater.
In chemistry , this equation can stimulate the air pollution in environmental cases.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
4
Pollutant Transport-Chemistry Models
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
5
Chemo-Taxis Problems from Mathematical Biology Like bacterial growth, tumor growth
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
6
Runge-Kutta Method Semi-discrete system Discretization of spatial operator like ∂x and ∂xx First discretizing the spatial operators on a chosen space grid, then PDE is converted into a system of ODEs. Then we use time integration method to obtain the fully discrete numerical solution
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
7
Time Integral Method:
quadrature rule General formula of Runge-Kutta Method:
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
8
Explicit if
The internal approximations can be computed one after another from an explicit relation
Implicit if else The must be retrieved from a system of linear or nonlinear algebraic relation, usually by a Newton type iteration.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
9
Runge-Kutta Method is often represented as Butcher-array
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
10
classical fourth-order explicit here p=s
the 2-stage Gauss method of order four It’s costly but better, because of the superior stability properties.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
11
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
12
The equation is like The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth. illustrate the effects of oscillations on problems Here I use 4th Order Explicit Runge-Kutta Method Let And the result is like
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
13
By Richardson extrapolate
With different step size here ratio=10 P=4.426
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
14
Let’s introduce the scalar, complex test equation
Let and application of rk equation to this test equation, we get , R is the stability function and here the function is to be
And z, b, A are the coefficients for butcher array Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
15
The stability function of an explicit method with p=s≤4 is given by the polynomial
The stability regions S for the stability function of degree s=1,2,3,4
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
16
A-stable means the stability regions S contains the left half-plane The exponential function also satisfies
L-stable is A-stable additional Gauss Method are A-stable and even L-stable
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
17
For the advection problem The step size restrictions is called CFL conditions Which formulated in Courant number
S=1
S=2
S=3
S=4
1st Upwind
1.00
1.00
1.25
1.39
2nd central
0.00
0.00
1.73
2.82
3rd upwindbiased
0.00
0.87
1.62
1.74
4th central
0.00
0.00
1.26
2.05
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
18
Maximal Value
f for stability
S=1
S=2
S=3
S=4
2nd central
0.5
0.5
0.62
0.69
4th central
0.37
0.37
0.47
0.52
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
19
Give value of numerical parameters: omega = 1 Nt=100, Nx=50, final time=1
We compare different initial value and different epsilon Red line for initial value Black line for explicit method Pink circle for implicit method
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
20
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
21
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
22
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
23
For initial value
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
24
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
25
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
26
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
27
Implicit method has much better stability properties.
The choices of numerical parameters are very important , for stability restrictions of advection, diffusion have special conditions.
Sufficiently fine grid can eliminate the troublesome oscillations. But very expensive.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
28
Thank you very much for you time and bye 谢谢!!
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
29
I.
Introduction
II.
Runge-Kutta Methods for ODE Systems
III.
Stability Analysis for the AdvectionDiffusion-Reaction Equation
IV.
Numerical Results
V.
Conclusions Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
2
The parameter a is advection velocity
is diffusion coefficient is source term coefficient Reaction term is logistic growth
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
3
Arise in many chemical and biological settings.
In hydrology , equations of this type model the transport and fate of adsorbing contaminants and microbe-nutrient systems in groundwater.
In chemistry , this equation can stimulate the air pollution in environmental cases.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
4
Pollutant Transport-Chemistry Models
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
5
Chemo-Taxis Problems from Mathematical Biology Like bacterial growth, tumor growth
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
6
Runge-Kutta Method Semi-discrete system Discretization of spatial operator like ∂x and ∂xx First discretizing the spatial operators on a chosen space grid, then PDE is converted into a system of ODEs. Then we use time integration method to obtain the fully discrete numerical solution
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
7
Time Integral Method:
quadrature rule General formula of Runge-Kutta Method:
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
8
Explicit if
The internal approximations can be computed one after another from an explicit relation
Implicit if else The must be retrieved from a system of linear or nonlinear algebraic relation, usually by a Newton type iteration.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
9
Runge-Kutta Method is often represented as Butcher-array
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
10
classical fourth-order explicit here p=s
the 2-stage Gauss method of order four It’s costly but better, because of the superior stability properties.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
11
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
12
The equation is like The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth. illustrate the effects of oscillations on problems Here I use 4th Order Explicit Runge-Kutta Method Let And the result is like
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
13
By Richardson extrapolate
With different step size here ratio=10 P=4.426
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
14
Let’s introduce the scalar, complex test equation
Let and application of rk equation to this test equation, we get , R is the stability function and here the function is to be
And z, b, A are the coefficients for butcher array Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
15
The stability function of an explicit method with p=s≤4 is given by the polynomial
The stability regions S for the stability function of degree s=1,2,3,4
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
16
A-stable means the stability regions S contains the left half-plane The exponential function also satisfies
L-stable is A-stable additional Gauss Method are A-stable and even L-stable
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
17
For the advection problem The step size restrictions is called CFL conditions Which formulated in Courant number
S=1
S=2
S=3
S=4
1st Upwind
1.00
1.00
1.25
1.39
2nd central
0.00
0.00
1.73
2.82
3rd upwindbiased
0.00
0.87
1.62
1.74
4th central
0.00
0.00
1.26
2.05
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
18
Maximal Value
f for stability
S=1
S=2
S=3
S=4
2nd central
0.5
0.5
0.62
0.69
4th central
0.37
0.37
0.47
0.52
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
19
Give value of numerical parameters: omega = 1 Nt=100, Nx=50, final time=1
We compare different initial value and different epsilon Red line for initial value Black line for explicit method Pink circle for implicit method
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
20
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
21
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
22
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
23
For initial value
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
24
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
25
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
26
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
27
Implicit method has much better stability properties.
The choices of numerical parameters are very important , for stability restrictions of advection, diffusion have special conditions.
Sufficiently fine grid can eliminate the troublesome oscillations. But very expensive.
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
28
Thank you very much for you time and bye 谢谢!!
Runge-Kutta Method for AdvectionDiffusion-Reaction Equation
9/17/2008
29
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