Numerical methods for ODEs

1. Numerical methods for ODEs Ordinary differential equations (ODEs) Dynamical system Widely applications in science & engineering Mixture problem Population models Point-mass motion Mechanical vibration Electrical networks Pendulum motion, �.

2. ODEs Different types First order ODE First order ODEs system High order ODE High order ODEs system Solutions Existence & uniqueness Analytical solution Numerical solution

3. Different numerical methods For first order ODE Single-step method Euler, Trapezoidal method Runge-Kutta methods: RK4 � most popular Stability and convergence Time-splitting (split-step) method Integration factor method Multi-step methods

4. Different numerical methods For first order ODEs system Direct extension of all methods for first order ODE For high order ODE First order ODEs system Direct discretization For high order ODE system First order ODEs system direct discretization

5. For first order ODE Consider initial-value problem (IVP): Choose a time step and partition [a,b] into N pieces Denote

6. Basic numerical methods Ideas: Derivative is the limit of difference Forward Euler method -- explicit method Backward Euler method � implicit method

7. Basic numerical methods Trapezoidal method � sum of forward & backward Euler methods Taylor expansion Forward Euler method Backward Euler method

8. Numerical example The problem Time step Forward Euler method Backward Euler method Trapezoidal method

9. Numerical results Example:

10. Order of accuracy Local truncation error: Def: The error at the given step if it is assumed that the previous results are all exact!! Ways to find: Replace in the difference equation Define the local truncation error Do Taylor expansion & use the ODE find the leading order term Order of accuracy: p

11. Order of accuracy Forward Euler method Local truncation error Do Taylor expansion (see details in class) Order of accuracy: first order & explicit Backward Euler method (exercise) Order of accuracy: first order & implicit Trapezoidal method (exercise) Order or accuracy: second order & implicit

12. Runge-Kutta methods Ideas Integrate the ODE over the interval Choose r points in the interval Evaluate slopes at these points via (forward) Euler method

13. Runge-Kutta methods Construct the averaged slope Determine the parameters via Taylor expansion such that the difference method has a given accuracy!!

14. Runge-Kutta methods Forward Euler method -- RK1 Backward Euler method Trapezoidal method

15. Runge-Kutta methods Second order Runge-Kutta method -- RK2 Choose Construct slopes Averaged slope RK2 method Determine the three parameters via Taylor expansion (see details in class or exercise)

16. Runge-Kutta methods RK2 (modified Euler method): Order of accuracy: 2 & explicit RK2 (modified midpoint method): Order of accuracy: 2 & explicit

17. Runge-Kutta methods Fourth order Runge-Kutta method -- RK4 Choose Construct slopes Averaged slope RK4 method Determine the seven parameters via Taylor expansion (see details in class or exercise)

18. Runge-Kutta methods 4th Order Runge-Kutta method -- most popular!! --- exercise!! Local Truncation error: Order of accuracy: 4 Explicit method

19. Numerical example The problem Exact Solution Different numerical methods Forward Euler method (FEM) Modified Euler method (MEM) 4th order Runge-Kutta method (RK4)

22. Numerical results Example:

23. Numerical results Population of fruitflies:

24. Numerical results Population of fruitflies:

25. ODE demo Some very interesting ODE demo sites: Interactive Differential Equations http://www.aw-bc.com/ide/ More