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Math 228B Theoretical Assignment 2 Winter 1999

Due: Wendesday, February 17, 1999

Problem 1   Use the von Neumann stability analysis to analyze the stability of the Lax-Friedrichs method,

$$u^{n+1}_j = { 1 \over 2 } \: ( u^n_{j-1} + u^n_{j+1} ) \: + \: {\sigma \over 2} \: ( u^n_{j-1} - u^n_{j+1} )$$

Problem 2   Prove Parseval's equality:

$$\vert\vert{\bf u}\vert\vert^2_2 \:=\: \vert\vert\widehat{\bf u}\vert\vert^2_2 ,$$

or equivalently

$$\frac{1}{M} \sum^{M-1}_{j=0} u_{j}^{2} \:=\:\sum_{k=-{M/2}+1} ^{{M/2}} b_k\:\overline{b_k}$$

where ${\bf u}=(u_0,\cdots,u_{M-1})$.

Problem 3   Use the von Neumann stability analysis to analyze the stability of LW4, the Lax-Wendroff method with fourth order spatial accuracy as defined by equations (103) and (106) on pages 26-27 in the notes.

Problem 4   Finish the proof that Lax-Wendroff is stable for all $\beta$. In other words, show that

$$\vert\lambda(\beta)\vert^2 = 1+(\sigma^4-\sigma^2)(1-cos\beta)^2 \leq 1 \quad for \: all \: \: \: 0 < \sigma \leq 1.$$

Problem 5   See if Lax-Wendroff can be derived in the same manner as Fromm. That is, given the discrete solution ${\bf u}^n =(u^n_0,\cdots,u^n_{M-1})$ at time tⁿ,

1.

Define an interpolating polynomial pⁿ_j(x) on each interval [x_j-1/2, x_j+1/2].

2.

Solve equation (1a,b)

$$\begin{eqnarray*}(1a)& \qquad \qquad \qquad u_t + a u_x &= 0 \\ (1b)& \qquad \qquad \qquad u(x, 0) &= \sum_j p^n_{j}(x) \: \chi_j(x) \\ \end{eqnarray*}$$

for one time step exactly $\Delta t$ using these the polynomials pⁿ_j(x) as initial data. (Recall that $\chi_j (x)$ is the characteristic function associated with the jth interval.)

3.

Integrating the exact solution found in Step 2 above to obtain uⁿ⁺¹_j, the average value of $u(x, \Delta {t})$ in the jth cell at the new time step.

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