runge-kutta method for.ppt

Jie, LiuCASA Seminar September 17 2oo8

I. Introduction

II. Runge-Kutta Methods for ODE Systems

III. Stability Analysis for the Advection-Diffusion-Reaction Equation

IV. Numerical Results

V. Conclusions

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Diffusion-Reaction Equation 2

The parameter a is advection velocity is diffusion coefficient is source term coefficient Reaction term is logistic growth

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Diffusion-Reaction Equation 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.

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Diffusion-Reaction Equation 4

Pollutant Transport-Chemistry Models

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Diffusion-Reaction Equation 5

Chemo-Taxis Problems from Mathematical Biology

Like bacterial growth, tumor growth

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Diffusion-Reaction Equation 6

Runge-Kutta Method Semi-discrete system Discretization of spatial operator like ∂x and

∂xxFirst 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

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Diffusion-Reaction Equation 7

Time Integral Method:

quadrature rule

General formula of Runge-Kutta Method:

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Diffusion-Reaction Equation 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.

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Diffusion-Reaction Equation 9

Runge-Kutta Method is often represented as Butcher-array

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Diffusion-Reaction Equation 10

classical fourth-order explicithere p=s

the 2-stage Gauss method of order fourIt’s costly but better, because of the superior stability properties.

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Diffusion-Reaction Equation 11

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 13

By Richardson extrapolate

With different step size

here ratio=10 P=4.426

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 17

For the advection problemThe step size restrictions is called CFL

conditionsWhich 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

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 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

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Diffusion-Reaction Equation 20

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Diffusion-Reaction Equation 21

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Diffusion-Reaction Equation 22

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Diffusion-Reaction Equation 23

For initial value

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Diffusion-Reaction Equation 24

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Diffusion-Reaction Equation 25

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Diffusion-Reaction Equation 26

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Diffusion-Reaction Equation 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.

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Diffusion-Reaction Equation 28

Thank you very much for you time and bye

谢谢 !!

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