Question

Presume that the problem is a Neumann type of heat equation. Determine the explicit matrix formulation of the Crank-Nicholson Scheme, that is, Bu ⱼ₊₁=Cuⱼ+bⱼ, see the notes documents. Explicitly write out the matrices B, C, and vector b of boundary, source, and/or initial condition data for any generic initial value function f(x). Display the actual B, C, and the iteration matrix A=B₋₁ C, determined using Matlab, for the case that n=10 ( n the number of spatial intervals). Then determine the condition number and norm of the iteration matrix A=B₋₁ C for the values n=10,20,100;r=0.1,0.6. Do not display A, B, or C when n=20,100. To display something, use the contour command Matlab to display a "contour plot" of the matrices for the larger sizes, it will display the type of matrix you have if you have a special form.

Answer 1

The Crank-Nicholson scheme is a numerical method used to solve the Neumann type of heat equation. It involves using matrices and vectors to represent the boundary, source, and initial condition data. The iteration matrix A can be determined by multiplying the inverse of matrix B with matrix C.

Step-by-step explanation:

The Crank-Nicholson scheme is a numerical method used to solve the Neumann type of heat equation. In this scheme, the equation can be written as Buj+1 = Cuj + bj, where B, C, and b are matrices and vectors containing the boundary, source, and initial condition data. To determine the explicit matrix formulation of the Crank-Nicholson scheme, we need to explicitly write out the matrices B, C, and vector b. The iteration matrix A is given by A = B-1C.