Computational Physics Lectures: Partial differential equations

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Explicit Scheme, solving the equations

Since all the discretized initial values

$\begin{equation*} u_{i,0} = g(x_i), \end{equation*}$ are known, then after one time-step the only unknown quantity is

$u_{i,1}$ which is given by

$\begin{equation*} u_{i,1}= \alpha u_{i-1,0}+(1-2\alpha)u_{i,0}+\alpha u_{i+1,0}= \alpha g(x_{i-1})+(1-2\alpha)g(x_{i})+\alpha g(x_{i+1}). \end{equation*}$ We can then obtain

$u_{i,2}$ using the previously calculated values

$u_{i,1}$ and the boundary conditions

$a(t)$ and

$b(t)$ . This algorithm results in a so-called explicit scheme, since the next functions

$u_{i,j+1}$ are explicitely given by Eq. (7).