Math 6620 • Analysis of Numerical Methods II

\begin{enumerate}

\item
Consider the initial value problem for the constant-coefficient diffusion equation (with \(\beta > 0\))
\[
v_t = \beta v_{xx},\quad t > 0
\]


with initial data \(v(x,0) = f(x)\). A scheme for this problem is:
\[
\frac{u_j^{n+1}-u_j^n}{k}
= \frac{\beta}{h^2}\left\{u_{j-1}^{n+1}-2u_j^{n+1}+u_{j+1}^{n+1}\right\}.
\]

Analyze the 2-norm stability of this scheme. For which values of \(k > 0\) and \(h > 0\) is the scheme
stable? (Note that there are no boundary conditions here.)


\item
Consider the variable coefficient diffusion equation
\[
v_t = (\beta(x)v_x)_x,\quad 0 < x < 1,\quad t > 0
\]


with Dirichlet boundary conditions
\[
v(0,t) = 0,\qquad v(1,t) = 0
\]


and initial data \(v(x,0) = f(x)\). Assume that \(\beta(x) \ge \beta_0 > 0\), and that \(\beta(x)\) is smooth. Let
\(\beta_{j+1/2} = \beta(x_{j+1/2})\). A scheme for this problem is:
\[
\frac{u_j^{n+1}-u_j^n}{k}
= \frac{1}{h^2}\left\{\beta_{j-1/2}u_{j-1}^{n+1}
-(\beta_{j-1/2}+\beta_{j+1/2})u_j^{n+1}
+\beta_{j+1/2}u_{j+1}^{n+1}\right\}.
\]


Analyze the 2-norm stability of this scheme for solving this initial boundary value problem. DO
NOT NEGLECT THE FACT THAT THERE ARE BOUNDARY CONDITIONS!

\end{enumerate}