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 Implicit Methods Adaptive Stepsize 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. Ode2: solves 1st and 2nd order ODE Ic1, ic2, bc: setting conditions ‘ do not evaluate Maxima on ODE

10. Linear ODE • Linearity: • Involves no product nor nonlinear functions of y and its derivatives • nth order linear ODE

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

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

14. Example

15. End of Review

16. The Canonical Problem This is Euler’s method

17. Example Compare with exact sol:

18. y 1 y=e–x x Example (cont)

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

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

21. 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)

22. Example (Euler’s)

23. Methods Improving Euler Motivated by Geometric Interpretation

24. Midpoint Method

25. Example (Midpoint)

26. Heun’s Method

27. Example (Heun’s) Note the result is the same as Midpoint!?

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

29. RK Methods

30. RK Methods (cont)

31. Taylor’s Expansion

32. RK 1st Order

33. RK 2nd Order

34. RK 2nd Order (cont)

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

36. Classical 4th order RK

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

38. 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

39. Example (cont)

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

41. Stability: Symptom

42. Symptom: Unstable Spring System Become unstable instantly … Start with this … Cause by stiff (k=4000) springs

43. Example Problem: Stability (cont) Conditionally stable

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

45. Review: Numerical Differentiation Taylor’s expansion: Forward difference Backward difference

46. Numerical Difference (cont) Central difference

47. Implicit Method (Backward Euler) Forward difference Backward difference

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

49. Stability limit Stiff Set of ODE Use the change of variable Get the following solution: A stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small