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\noindent{\bf Math 269B} {\bf Homework \#4} Due date: Friday, February 17
\noindent Instructor: Luminita Vese. Teaching Assistant: Michael Puthawala.
\bigbreak
\noindent{\bf [1]} Consider the one-way wave equation
$$\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0.$$
Analyze the stability and the order of accuracy of the following {\it angled
derivative method}
$$u^{n+2}_j=(1-2\lambda)(u^{n+1}_j-u^{n+1}_{j-1})+u^n_{j-1},\ \ \ n\geq0,$$
with $\lambda=\frac{\triangle t}{\triangle x}$.
\bigbreak
(Reminder to me: CHECK B.C. for the computational assignment)
\noindent{\bf [2]} {\it Computational assignment.} Consider the one-way wave equation
\begin{eqnarray*}
& &\frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0,\ \ t>0, \ \ 00$ (positive speed of propagation).
\bigbreak
\noindent{\bf [4]} Consider the nonlinear equation $u_t+u_x=\cos^2u$, approximated by the
Lax-Wendroff scheme with $R_{k,h}f_m^n=f_m^n$, treating the $\cos^2u$ term as
$f(t,x)$. Show that the obtained scheme is first order accurate (use
$\lambda=\frac{k}{h}$).
\bigbreak
\noindent{\bf [5]} Consider the box scheme
\begin{eqnarray*}
\frac{1}{2k}[(v_m^{n+1}+v_{m+1}^{n+1})-(v_m^{n}+v_{m+1}^{n})]
\\
+\frac{a}{2h}[(v_{m+1}^{n+1}-v_{m}^{n+1})+(v_{m+1}^{n}-v_{m}^{n})]\\
=\frac{1}{4}(f_{m+1}^{n+1}+f_{m}^{n+1}+f_{m+1}^n+f_m^n)
\end{eqnarray*}
(a) Show that the scheme is an approximation to the one-way wave equation
$u_t+au_x=f$ that is accurate
of order (2,2).
(b) Show that the scheme is stable for all values of $\lambda$.
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