MATH 661.05 Numerical Solution of Differential Equations

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MATH 661.05 (Numerical Solution of Differential Equations) - Fall 2021

Assignment #5

1. Consider the following 2D heat equation on and

where the true solution is

For both parts (a) and (b), you can write your own codes, or modify the provided Matlab programs (pay attention to the places with ???? in the code).

(a) Solve this initial-boundary value problem using FTCS (forward time central space) algorithm. Verify that FTCS is first-order in time and second-order in space. When verify the order of convergence in space, you can fix the time stepsize ∆t very small. When verify the order of convergence in time, keep in mind that the stability condition should be satisfied.

(b) Solve this initial-boundary value problem using Crank-Nicolson algorithm. Ver-ify that Crank-Nicolson algorithm is second-order in both time and space. When verify the order of convergence in space, you can fix the time stepsize (∆t) at very small. When verify the order of convergence in time, fix the spatial stepsize (∆x, ∆y) small.

2. Use central finite difference method to solve the 2D acoustic wave equation:

where the true solution is given by

The two initial conditions (if the time stepsize is ) are given by

(a) First run your code for record the maximal error at t = 1, then run your code for verify that the method is second-order accuracy in space by comparing the errors.

(b) Recall that the stability for the central finite difference method is Here fix c = 1 and choose a value of so that the stability is violated, then run your code to see if it is convergent.

3. Consider the model problem

Verify that the algebraic system from the linear Galerkin method is equivalent to that of finite difference method when the mesh is uniform, i.e.