1. Lecture 34 - Ordinary Differential Equations - BVP CVEN 302 November 28, 2001

2. Systems of Ordinary Differential Equations - BVP Shooting Method for Nonlinear BVP Finite Difference Method Partial Differential Equations

3. Shooting Method for Nonlinear ODE-BVPs Nonlinear ODE Consider with guessed slope t Use the difference between u(b) and y b to adjust u’(a) m(t) = u(b, t) - y b is a function of the guessed value t Use secant method or Newton method to find the correct t value with m(t) = 0

4. Nonlinear Shooting Based on Secant Method Nonlinear ODE

5. MATLAB Example in Nonlinear Shooting Method Nonlinear shooting with secant method Convert to two first-order ODE-IVPs Update t using the secant method

6. Nonlinear Shooting - Secant Method y(x) y’(x)

7. Nonlinear Shooting Based on Newton’s Method Nonlinear ODE Check for convergence of m(t)

8. Nonlinear Shooting Based on Newton’s Method Nonlinear ODE-IVP Chain Rule x and t are independent 0

9. Nonlinear Shooting with Newton’s Method Solve ODE-IVP Construct the auxiliary equations

10. Nonlinear Shooting with Newton’s Method Calculate m(t) -- deviation from the exact BC Update t by Newton’s method

11. Finite-Difference Methods Divide the interval of interest into subintervals Replace the derivatives by appropriate finite-difference approximations in Chapter 11 Solve the system of algebraic equations by methods in Chapters 3 and 4 For nonlinear ODEs, methods in Chapter 5 may be used

12. Finite-Difference Method General Two-Point BVPs Replace the derivatives by appropriate finite-difference approximations xixi x i-1 x i+1 hhhh

13. Finite-Difference Method Central difference approximations Tridiagonal system

14. Finite-Difference Method Central Difference ==> Tridiagonal system

15. Finite-Difference Method for Nonlinear BVPs Nonlinear ODE-BVPs Evaluate f i by appropriate finite-difference approximations xixi x i-1 x i+1 hhhh

16. Finite-Difference Method for Nonlinear BVPs SOR method Iterative solution Convergence criterion

17. Example 14.12 - MATLAB Note error in Text f i : negative sign

18. Chapter 15 Partial Differential Equations

19. Classification of PDEs General form of linear second-order PDEs with two independent variables linear PDEs: a, b, c,….,g = f(x,y) only

20. Heat Equation: Parabolic PDE Heat transfer in a one-dimensional rod x = 0x = a g 1 (t)g 2 (t)

21. Discretize the solution domain in space and time with h =  x and k =  t Time (j index) space (i index) x t

22. Initial and Boundary Conditions Initial conditions : u(x,0) = f(x) u(0, t) = g 1 (t) u(a, t) = g 2 (t) Explicit Euler method

23. Heat Equation Finite-difference (i,j)(i+1,j)(i-1,j) (i,j+1) u(x,t) x x t t xixi x i+1 x i-1 tjtj t j+1 Forward-difference Central-difference at time level j

24. Explicit Method Explicit Euler method for heat equation Rearrange Stability:

25. Explicit Euler Method Stable Unstable (negative coefficients)

26. Heat Equation: Explicit Euler Method r = 0.5

27. Example: Explicit Euler Method Heat Equation (Parabolic PDE) c = 0.5, h = 0.25, k = 0.05 12340 20 + 40 x 60e -2t 20e -t 0 1 2

28. Example Explicit Euler method First step: t = 0.05

29. Second step: t = 0.10 12340 20 + 40 x 60e -2t 20e -t 304050 29.614047.72

30. Heat Equation: Time-dependent BCs r = 0.4

31. Stability for Explicit Euler Method It can be shown by Von Neumann analysis that Switch to Implicit method to avoid instability Numerical Stability

32. Explicit Euler Method: Stability Unstable !! r = 1

33. Implicit Euler method Initial conditions : u(x,0) = f(x) u(0, t) = g 1 (t) u(a, t) = g 2 (t) Unconditionally Stable

34. Implicit Method Finite-difference (i,j) (i+1,j+1)(i-1,j+1)(i,j+1) T(x,t) x x t t xixi x i+1 x i-1 tjtj t j+1 Forward-difference Central-difference at time level j+1

35. Implicit Euler Method Implicit Euler method for heat equation Tridiagonal matrix (Thomas algorithm) Unconditionally stable

36. Implicit Euler Method Unconditionally stable r = 2

37. Example: Implicit Euler Method Heat Equation (Parabolic PDE) c = 0.5, h = 0.25, k = 0.1 12340 20 + 40 x 60e -2t 20e -t 0 1

38. Example Implicit Euler method

39. Solve the tridiagonal matrix 12340 20 + 40 x 60e -2t 20e -t 0 1 28.9638.5146.19

40. Crank-Nicolson method Initial conditions : u(x,0) = f(x) u(0, t) = g 1 (t) u(a, t) = g 2 (t) Implicit Euler method : first-order in time Crank-Nicolson : second-order in time

41. Crank-Nicolson Method Crank-Nicolson method for heat equation Average between two time levels Tridiagonal matrix Unconditionally stable (neutrally stable) Oscillation may occur

42. General Two-Level Method General two-stage method for heat equation Weighted-average of spatial derivatives between two time levels n and n+1

43. Example: Crank-Nicolson Method Heat Equation (Parabolic PDE) c = 0.5, h = 0.25, k = 0.1 12340 20 + 40 x 60e -2t 20e -t 0 1

44. Example Crank-Nicolson method Tridiagonal matrix (r = 0.8)

45. Solve the tridiagonal matrix 12340 20 + 40 x 60e -2t 20e -t 0 1 29.4239.3047.43

46. Implicit Euler method Unconditionally stable r = 2

47. Heat Equation with Insulated Boundary No heat flux at x = 0 and x = a x = 0x = a u x (a,t) = 0 u x (0,t) = 0

48. Insulated Boundary No heat flux at x = a x = a x n+1 x n-1 xnxn u x (a,t)=0