Math 228B Final Homework Assignment Winter 1999

Problem 1 Let

$\begin{displaymath}\Lambda ( \beta ) = \sum_s c_s \, e^{i \beta s} \end{displaymath}$

be the symbol of a consistent¹ linear finite difference operator L. Show that the following statements are true.

1.

The Taylor series expansion for $\Lambda ( \beta )$ begins with a 1,

$\begin{displaymath}\Lambda ( \beta ) = \, 1 \, + ... \end{displaymath}$

2.

$\begin{displaymath}\sum_s c_s \, = 1 \, . \end{displaymath}$

3.

$\begin{displaymath}\sum_s c_s \, s\, = - \sigma \, . \end{displaymath}$

Problem 2 Recall the following definitions:

Definition 1: A linear finite difference method u_jⁿ⁺¹ = L ( uⁿ )_j is called monotonicity preserving if and only if

$\begin{displaymath}u_j^n \ge u_{j+1}^n \quad \hbox{for all j} \quad {\, \Right... ...\quad u_j^{n+1} \ge u_{j+1}^{n+1} \quad \hbox{for all j} \, . \end{displaymath}$

Definition 2: A linear finite difference method is called positivity preserving if and only if

$\begin{displaymath}u_j^n \, > \,0 \quad \hbox{for all j} \quad {\, \Rightarrow \,}\quad u_j^{n+1} \, > \,0 \quad \hbox{for all j} \, . \end{displaymath}$

Definition 3: A linear finite difference method is called max norm bounded if and only if

$\begin{displaymath}{\vert \! \vert}L u^n {\vert \! \vert}_{\infty} \le {\vert \! \vert}u^n {\vert \! \vert}_{\infty} \end{displaymath}$

Show that these three conditions are equivalent; i.e., show that

$\begin{eqnarray*}\hbox {L {\rm is monotonicity preserving}} &{\, \Rightarrow \... ...{\, \Rightarrow \,}&\hbox {L {\rm is monotonicity preserving}} \end{eqnarray*}$

or any equivalent permutation of these statements.

Problem 3 Modified Equation Analysis

1.

Do a complete modified equation analysis of the Lax-Wendroff (LW2) and Beam-Warming methods. I want you to develop the dispersion relations and compute the wave speed for each method. The Beam Warming method is

$\begin{displaymath}u^{n+1}_j = u^n_j - {\sigma \over 2} \, ( 3 u^n_j - 4 u^n_{j-... ... + {\sigma^2 \over 2} \, ( u^n_j - 2 u^n_{j-1} + u^n_{j-2} ) \end{displaymath}$

2.

Now use LW2 and Beam-Warming to compute the numerical solution to the standard linear advection equation with the discontinuous (shock wave) initial data up to time T₀ for some reasonable T₀, such as T₀ = 1 or T₀ = 1.

3.

Graph these solutions against the exact solution and use the modified equation analysis to explain what you see.

Footnotes