Ordinary Differential Equations

1. Ordinary Differential Equations Jyun-Ming Chen

2. Review Euler’s method 2nd order methods Midpoint Heun’s Runge-Kutta Method Systems of ODE Stability Issue Contents

3. Review • DE (Differential Equation) • An equation specifying the relations among the rate change (derivatives) of variables • ODE (Ordinary DE) vs. PDE (Partial DE) • The number of independent variables involved

4. Solution of an equation: Solution of DE vs. Solution of Equation f(x) x Review (cont) • Geometrically,

5. Solution of an differential equation: Geometrically: x t Need additional conditions to specify a solution Review (cont)

6. Review (cont) • Order of an ODE • The highest derivative in the equation • nth order ODE requires n conditions to specify the solution • IVP (initial value problem): All conditions specified at the same (initial) point • BVP (boundary value problem): otherwise

7. IVP VS. BVPRevisit Shooting Problem

8. IVP vs. BVP Physical meaning

9. Review (cont) • Linearity: • No product nor nonlinear functions of y and its derivatives • nth order linear ODE

10. Focus of This Chapter • Solve IVP of nth order ODE numerically • e.g.,

11. ODE (IVP) • First order ODE (canonical form) • Every nth order ODE can be converted to n first order ODEs in the following method:

13. Example

14. End of Review

15. The Canonical Problem This is Euler’s method

16. Example

17. y 1 y=e–x x Example (cont)

18. Error Analysis(Geometric Interpretation) Think in terms of Taylor’s expansion If the true solution were a straight line, then Euler is exact

19. Error Analysis(From Taylor’s Expansion) Euler’s Euler’s truncation error O(Dx2) per step 1st order method

20. y Cumulative Error x Remark: Dx Error  But computation time x = 0 x = T Number of steps = T/Dx Cumulative Err. = (T/Dx)  O(Dx2) = O(Dx)

21. Example (Euler’s)

22. Methods to Improve Euler Motivated by Geometric Interpretation

23. Midpoint Method

24. Example (Midpoint)

25. Heun’s Method

26. Example (Heun’s)

27. Comparison of Euler, Heun, midpoint 1st order: Euler 2nd order: Heun, midpoint “order”: All are special cases of RK (Runge-Kutta) methods Remark

28. RK Methods

29. RK Methods (cont)

30. Taylor’s Expansion

31. RK 1st Order

32. RK 2nd Order

33. RK 2nd Order (cont)

34. RK 4th Order • Mostly commonly used one • Higher order … more evaluation, but less gain on accuracy

35. System of ODE • Convert higher order ODE to 1st order ODEs • All methods equally apply, in vector form

36. Initial Condition c m k x Example (Mass-Spring-Damper System) • Governing Equation • After setting the initial conditions x(0) and x’(0), compute the position and velocity of the mass for any t > 0

37. Example (cont)

38. Assume m=1,c=1, k=1 (for ease of computation) Example (cont) set Dt=0.1

39. Stability: Symptom

40. Example Problem: Stability (cont) Conditionally stable

41. Discussion • Different algorithm different stability limit • Check Midpoint Method • Different problem different stability limit • use the previous problem as benchmark

42. Implicit Method (Backward Euler)

43. Example • Remark: • Always stable (for this problem) • Truncation error the same as Euler (only improve the stability)

44. Linear System of ODE with Constant Coefficients

45. Semi-Implicit Euler • Not guaranteed to be stable, but usually is

46. Initial Condition c m k x

47. Initial Condition c m k x

48. Stability limit Stiff Set of ODE Use the change of variable Get the following solution:

50. Solving ODE numerically … tracing the integral curve y(x) what’s wrong with uniform step size Uniformly small: waste effort Uniformly large: might miss details Adaptive Stepsize