1. Ordinary Differential Equations
Jyun-Ming Chen

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

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

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

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. BVP Revisit Shooting Problem

8. IVP vs. BVP
Physical meaning

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

10. Linear ODE Linearity: nth order linear ODE
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. Example (cont) 1 x y
y=e–x

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

28. Remark Comparison of Euler, Heun, midpoint “order”:
1st order: Euler
2nd order: Heun, midpoint
“order”:
All are special cases of RK (Runge-Kutta) methods
Exact
Euler (error)
Midpoint (error)
y(0.1)
0.905
0.9 (0.005)
(0)
y(0.2)
0.819
0.81 (0.009)
y(0.3)
0.741
0.729 (0.012)

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. 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 c k
Initial Condition x m

39. Example (cont)

40. Example (cont) set Dt=0.1 Assume m=1,c=1, k=1
(for ease of computation)
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. Stability (cont)
Example Problem:
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. Stiff Set of ODE Use the change of variable
Get the following solution:
Stability limit
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

51. Linear System of ODE with Constant Coefficients
Do not really use (..)-1. Solve linear system instead

52. Analysis Explicit and implicit Euler both in the form:
represent x(0) in eigen basis
x will converge if |li|  1

53. Analysis (cont) Stiff equations have large eigenvalues
Explicit Euler requires small h to converge
Implicit Euler always converges (in this problem)

54. Example
Explicit
Implicit

55. Semi-Implicit Euler Not guaranteed to be stable, but usually is
Solving implicit methods by linearization is called a “semi-implicit” method
Semi-Implicit Euler
Not guaranteed to be stable, but usually is
Jacobian

56. About Jacobian
Taylor’s expansion:
Jacobian

57. c k
Initial Condition x m
Implicit Euler

58. c k
Initial Condition x m
Semi-Implicit Euler

59. Ex: Semi-Implicit Euler

60. Amazingly, this translates to…
Very similar to Verlet integration formula… no wonder Verlet is pretty stable

61. Adaptive Stepsize
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

62. Step Doubling
Idea:
Estimate the truncation error by taking each step twice: one full step, two half steps
control the step size such that the estimated error is not too big.
1(2h1)
2(h1)
Desired h0

63. Ex: RK 2nd Order
Overhead:
# of f(x,y) evaluations
24–2 = 6

64. Adaptive Step with RK4 (NR)

65. GSL

66. Initialization

67. Iteration