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

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

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

Nonlinear Shooting Based on Secant Method Nonlinear ODE

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

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

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

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

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

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

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

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

Finite-Difference Method Central difference approximations Tridiagonal system

Finite-Difference Method Central Difference ==> Tridiagonal system

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

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

Example MATLAB Note error in Text f i : negative sign

Chapter 15 Partial Differential Equations

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

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

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

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

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

Explicit Method Explicit Euler method for heat equation Rearrange Stability:

Explicit Euler Method Stable Unstable (negative coefficients)

Heat Equation: Explicit Euler Method r = 0.5

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

Example Explicit Euler method First step: t = 0.05

Second step: t = x 60e -2t 20e -t

Heat Equation: Time-dependent BCs r = 0.4

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

Explicit Euler Method: Stability Unstable !! r = 1

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

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

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

Implicit Euler Method Unconditionally stable r = 2

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

Example Implicit Euler method

Solve the tridiagonal matrix x 60e -2t 20e -t

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

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

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

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

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

Solve the tridiagonal matrix x 60e -2t 20e -t

Implicit Euler method Unconditionally stable r = 2

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

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