重要

Consider the time-dependent Cauchy probelm of scalar equation,

$$\begin{equation} \left\{ \begin{array}{lr} u_t + au_x = 0, \qquad -\infty < x < \infty, t \geq 0 \\ u(x,0) = u_0(x) \end{array} \right. \end{equation}$$

Define:

• $h = \Delta x$
• $k = \Delta t$
• $x_j = jh, j = \cdots,-1,0,1,\cdots$
• $t_n = nk, n = 0,1,2,\cdots$
• $x_{j+\frac{1}{2}} = x_j + h/2 = (j+\frac{1}{2})h$
• $\lambda = \frac{k}{h}$

1D linear advection equation

$$\begin{equation} \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0, x_{L} \leq x \leq x_{R}, t\geq 0 \tag{1} \end{equation}$$

where the independent variables are $t$(time) and $x$(space)

• $x$ is restricted to the finite interval $[p,q]$ which is called the computational domain.
• $a$ is a constant and the denpend variable
• where $u$ = $u(x,t)$