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

9/17/2008Runge-Kutta Method for Advection-

Diffusion-Reaction Equation 2

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



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.



Pollutant Transport-Chemistry Models



Chemo-Taxis Problems from Mathematical Biology

Like bacterial growth, tumor growth



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



Time Integral Method:

quadrature rule

General formula of Runge-Kutta Method:



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 is often represented as Butcher-array



classical fourth-order explicithere p=s

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





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



By Richardson extrapolate

With different step size

here ratio=10 P=4.426



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



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



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



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



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



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









For initial value









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.



Thank you very much for you time and bye

谢谢 !!



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Runge-Kutta Method for.ppt