Runge-Kutta Methods for Advection-Diffusion-Reaction Equations

1. Runge-Kutta Methods for Advection-Diffusion-ReactionEquations Jie, Liu CASA Seminar September 17 2oo8

2. Contents • Introduction • Runge-Kutta Methods for ODE Systems • Stability Analysis for the Advection-Diffusion-Reaction Equation • Numerical Results • Conclusions Runge-Kutta Method for Advection-Diffusion-Reaction Equation

3. 1 Introduction of our model problem The parameter a is advection velocity • is diffusion coefficient • is source term coefficient • Reaction term is logistic growth Runge-Kutta Method for Advection-Diffusion-Reaction Equation

4. Application of A-D-R Equations • 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 Advection-Diffusion-Reaction Equation

5. The Examples • Pollutant Transport-Chemistry Models Runge-Kutta Method for Advection-Diffusion-Reaction Equation

6. Chemo-Taxis Problems from Mathematical Biology • Like bacterial growth, tumor growth Runge-Kutta Method for Advection-Diffusion-Reaction Equation

7. 2 Runge-Kutta Methods for ODE Systems • 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 Advection-Diffusion-Reaction Equation

8. How Runge-Kutta Method for ODE System • Time Integral Method: • quadrature rule • General formula of Runge-Kutta Method: Runge-Kutta Method for Advection-Diffusion-Reaction Equation

9. Explicit and Implicit • 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 Advection-Diffusion-Reaction Equation

10. Butcher-Array • Runge-Kutta Method is often represented as Butcher-array Runge-Kutta Method for Advection-Diffusion-Reaction Equation

11. Comparison of Implicit and Explicit method 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 Advection-Diffusion-Reaction Equation

12. explicit implicit Runge-Kutta Method for Advection-Diffusion-Reaction Equation

13. Example: Logistic Equation • 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 Advection-Diffusion-Reaction Equation

14. Results • By Richardson extrapolate • With different step size • here ratio=10 • P=4.426 Runge-Kutta Method for Advection-Diffusion-Reaction Equation

15. The Stability Analysis for A-D-R Equation 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 Advection-Diffusion-Reaction Equation

16. 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 Advection-Diffusion-Reaction Equation

17. A-stable and L-stable • 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 Advection-Diffusion-Reaction Equation

18. Step Size Restrictions for Advection-Diffusion • For the advection problem The step size restrictions is called CFL conditions Which formulated in Courant number Runge-Kutta Method for Advection-Diffusion-Reaction Equation

19. Stability Restrictions for Diffusion Maximal Value f for stability Runge-Kutta Method for Advection-Diffusion-Reaction Equation

20. Numerical Results • 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 Advection-Diffusion-Reaction Equation

21. Effect of Reaction Runge-Kutta Method for Advection-Diffusion-Reaction Equation

22. Effect of Diffusion Runge-Kutta Method for Advection-Diffusion-Reaction Equation

23. Effect of Advection Runge-Kutta Method for Advection-Diffusion-Reaction Equation

24. Effect of step size selection • For initial value Runge-Kutta Method for Advection-Diffusion-Reaction Equation

25. Effect of different diffusion parameter Runge-Kutta Method for Advection-Diffusion-Reaction Equation

26. For another initial value Runge-Kutta Method for Advection-Diffusion-Reaction Equation

27. Effect of different advection parameters Runge-Kutta Method for Advection-Diffusion-Reaction Equation

28. Conclusion • 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 Advection-Diffusion-Reaction Equation

29. End • Thank you very much for you time and bye 谢谢!! Runge-Kutta Method for Advection-Diffusion-Reaction Equation