Math 6610 • Analysis of Numerical Methods I

{Heat Equation Stability:}

Consider the variable coefficient diffusion equation
\[
  v_t=(\beta v_x)_x,\qquad 0<x<1,\quad t>0
\]
with Dirichlet boundary conditions
\[
  v(0,t)=0,\qquad v(1,t)=0
\]
and initial data \(v(x,0)=f(x)\). Assume that \(\beta(x)\geq\beta_0>0\), 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.

NOTE: Because of the boundary conditions and the variable coefficients, von
Neumann (Fourier) analysis does not apply.

\textbf{7) Numerical Methods for ODEs:} Consider the initial value problem for
a system of ODEs
\[
  y'=Ay
\]
where \(A\) has eigenvalues with real parts ranging from \(-10^6\) to \(-1\). Which
of the following schemes would be the best choice for solving this problem?
Justify your answer in terms of stability, accuracy, and efficiency. Does your
answer change depending on the size of the system and the condition number of
\(A\)? Explain your answer.
\[
  y^{n+1}=y^n+kAy^n,
  \tag{1}
\]
\[
  y^{n+1}=y^n+kAy^{n+1},
  \tag{2}
\]
\[
  y^{n+1}=y^n+\frac{k}{2}\left(Ay^n+Ay^{n+1}\right).
  \tag{3}
\]