Computational Physics Lectures: Partial differential equations

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Explicit Scheme, boundary conditions

The boundary conditions are

$\begin{equation*} u(0,t)= a(t) \hspace{0.5cm} t \ge 0, \end{equation*}$ and

$\begin{equation*} u(L,t)= b(t) \hspace{0.5cm} t \ge 0, \end{equation*}$ where

$a(t)$ and

$b(t)$ are two functions which depend on time only, while

$g(x)$ depends only on the position

$x$ . Our next step is to find a numerical algorithm for solving this equation. Here we recur to our familiar equal-step methods and introduce different step lengths for the space-variable

$x$ and time

$t$ through the step length for

$x$

$\begin{equation*} \Delta x=\frac{1}{n+1} \end{equation*}$ and the time step length

$\Delta t$ . The position after

$i$ steps and time at time-step

$j$ are now given by

$\begin{equation*} \begin{array}{cc} t_j=j\Delta t & j \ge 0 \\ x_i=i\Delta x & 0 \le i \le n+1\end{array} \end{equation*}$