(PDF file: paper9.pdf)

3 Approximate Solution

In this section, we construct an approximate solution of the Lax-Friedrichs difference scheme type for the initial-boundary value problem (1), (2) in a standard way (cf. [3],[4],[14].)

Let the initial data $u(0,x)$ be a bounded measurable function and

$$\Vert u(0,\cdot)\Vert _{L^\infty} \leq M.$$

We suppose that the inverse of the $x$-mesh length $\Delta x$ and the ratio of the period $T$ and the $t$-mesh length are even integers $2L$ and $2N$,

$$2L\Delta x= 1, \hspace{2em}2N\Delta t=T.$$

It is necessary that the ratio of $\Delta x$ and $t$-mesh length $\Delta t$ is a sufficiently large constant for the CFL condition. We take the value such that

$$\frac{\Delta x}{\Delta t} \geq \Lambda \equiv\max_{\vert u\vert\leq M+TG_0}\vert f'(u)\vert,$$ (17)

where $T$ is time-period of function $g$, and $G_0$ is the maximum value of $\vert g\vert$

$$G_0 = \max_{[0,T]\times[0,1]}\vert g(t,x)\vert.$$

Let $E^n_j$ be an interval and $J_n$ an index set such that

$$\left\{\begin{array}{lll} E^n_j & = & ((j-1)\Delta x,(j+1)\... ... & \mbox{if $n$ is odd}, \end{array}\right.\end{array}\right.$$

and we denote $R(t,x;u_l,u_r)$ as the solution of the Riemann problem (16).

The approximation of the initial value $u^\Delta (0,x)$ is defined as a step value function

$$u^\Delta (0,x) = u^0_j \equiv \frac{1}{m(E^0_j)}\int_{E^0_j} u(0,x)dx \hspace{2em}\mbox{on } E^0_j \hspace{1em}(j\in J_0),$$

where $u(0,x)$ is extended to the function on ${\mbox{\sl R}}$ as the $x$-periodic function. The value $u^0_j$ has the periodicity of $u^0_{j+2L}=u^0_j$. We define $u^\Delta (t,x)$ as the solution of the Riemann problem for the step initial data $u^\Delta (0,x)$

$$u^\Delta (t,x)=R(t,x-j\Delta x,u^0_{j-1},u^0_{j+1})$$

in each small regions $(0,\Delta t)\times E^1_j$, $j\in J_1$. The wave must arrive at the top of the region by the CFL condition (17). On the line $t=\Delta t$, we define $u^\Delta (\Delta t,x)$ as the mean value

$$u^\Delta (\Delta t,x) = u^1_j \equiv \frac{1}{m(E^1_j)}\int... ...t-0,x)dx \hspace{2em}\mbox{on } E^1_j \hspace{1em}(j\in J_1),$$

and construct $u^\Delta (t,x)$ by solutions of Riemann problems in $(\Delta t,2\Delta t)\times{\mbox{\sl R}}$ similarly in $(0,\Delta t)\times{\mbox{\sl R}}$,

$$u^\Delta (t,x) = R(t-\Delta t,x-j\Delta x,u^1_{j-1},u^1_{j+1... ...x{in } (\Delta t,2\Delta t)\times E^2_j\hspace{1em}(j\in J_2).$$

On $t=2\Delta t$, we set $u^\Delta (2\Delta t,x)$ as the sum of the mean value and the term for the outer force

$$\begin{eqnarray*} \lefteqn{ u^\Delta (2\Delta t,x) = u^2_j = u^2_{0,j} + 2\Delt... ...^2_j} g(t,x)dx \\ & & \mbox{on } E^2_j \hspace{1em}(j\in J_2). \end{eqnarray*}$$

The last calculation is called the fractional step method.

For $(2n\Delta t,(2n+2)\Delta t]\times {\mbox{\sl R}}$ ( $n=1,2,\ldots,N-1$) we define the approximation $u^\Delta (t,x)$ by the similar way. The following lemma shows that above construction can be continued to $n=N-1$.

LEMMA 2   If constants $u_l$ and $u_r$ satisfy

$$\vert u_l\vert\leq A, \hspace{1em}\vert u_r\vert\leq A,$$

then the solution of the Riemann problem (16) $u(t,x)$ satisfies $\vert u(t,x)\vert\leq A$.

By Lemma 2, it is easy to show that $\vert u^\Delta (t,x)\vert\leq M+TG_0$ and waves appeared in the definition of $u^\Delta (t,x)$ cannot access in $(0,T)\times{\mbox{\sl R}}$ from the CFL condition (17) because speeds of these waves do not over the maximum value of $\vert f'(u)\vert$.