The one$-$dimensional advection$-$dispersion equation $($ADE$)$ without sink$/$source

is as follows

$\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{t}}=D\frac{de{l}^{2}C}{del{x}^2}-v\frac{\text{d}\text{e}\text{l}\text{C}}{\text{d}\text{e}\text{l}\text{x}},$

where $D=$ hydrodynamic dispersion coefficient $[\frac{L^2}{T}],V=$ groundwater flow

velocity $[\frac{L}{T}],c=$ concentration $[$M$/$L$3],x=$ spatial coordinate $[$L$],t=$ time $T.$

Assuming $v=0\mathrm{.}6\frac{m}{?}$ day, $D=5\frac{m^2}{?}$ day. The length of domain $L=50m$ is

discretized into $5$ segments using a step $x=10m.$ The time interval is $t=$

$1$ day.

The initial condition is: