91Ó°ÊÓ

Derive the truncation error for the centered difference approximation of the second derivative.

Describe a numerical scheme to solve Poisson's equation $$ \nabla^{2} u=f(x, y) \text {, } $$ (assuming that \(\Delta x=\Delta y\) ) analogous to: (a) Jacobi iteration (b) Gauss-Seidel iteration (c) \(S-\mathbf{O}-\mathrm{R}\)

Describe a numerical scheme (based on Jacobi iteration) to solve Laplace's equation in three dimensions. Estimate the number of iterations necessary to reduce the error in half.

Show that Jacobi iteration corresponds to the two-dimensional diffusion equation, by taking the limit as \(\Delta x=\Delta y \rightarrow 0\) and \(\Delta t \rightarrow 0\) in some appropriate way.

Using forward differences in time and centered differences in space, analyze carefully the stability of the difference scheme if the boundary condition for the heat equation is $$ \frac{\partial u}{\partial x}(0)=0 \quad \text { and } \quad \frac{\partial u}{\partial x}(L)=0 $$ (Hint: See Sec. 13.3.9) Compare your result to the one for the houndary conditions \(u(0)=0\) and \(u(L)=0\).