Forward Time Central Space - method

With the Forward Time Central Space - method (FTCS), the exact function $u(x,t)$ is approximated by a grid function $u_{h,k}(x_l,t_m)$, $x_l=l\cdot h$, $l=1,2,\ldots,n$ with $${h=\frac{L}{n+1}}$$ and $t_m=m\cdot k$, $m=0,1,2,\ldots$ .
With this grid function we implement a Carthesian Grid on the region
$\Omega=[0;L]\times [0;\infty)$ with $h$ as step size in space direction and step size $k$ in time direction.

$$u_{h,k}(x_l,t_m)\equiv u_l^m$$

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The Taylor Series for a function $u(x)$ enables an analysis of the function around a specific $x$ using $u(x)$ and derivatives of $u$:

$$u(x+h)=u(x)+h\cdot u'(x)+\frac{h^2}{2}u''(x)+ \frac{h^3}{6}+u'''(x)+\frac{h^4}{24}u^{IV}(\xi).$$ (28)

If we stop the Taylor Series right after the linear term we get an approximation for the first derivative:

$$u'(x)=\frac{u(x+h)-u(x)}{h}.$$ (29)

The error is proportional to the step size $h$.
Similar to $u(x+h)$ we can approximate $u(x-h)$:

$$u'(x)=\frac{u(x)-u(x-h)}{h}.$$ (30)

By subtracting (30) from (29) and dividing by $2h$, we obtain the symmetrical difference

$$\begin{eqnarray*} u'(x)=\frac{u(x+h)-u(x-h)}{2h}. \end{eqnarray*}$$

With the same idea we can get an approximation for the second derivative:

$$u''(x)=\frac{u(x+h)-2u(x)+u(x-h)}{h^2}.$$ (31)

Considered is again the Diffusion equation $u_t\;=\;Du_{xx}$:

$$\frac{\partial u(x,t)}{\partial t} = D\frac{\partial^2u(x,t)} {\partial x^2}.$$

By applying (29) and (31) on the Diffusion equation with respect to the Cartesian Grid values $u_l^m$ we obtain an approximation for $u_l^{m+1}$:

$$\frac{u_l^{m+1}-u_l^m}{k} = D\frac{u_{l-1}^m-2u_l^m+u_{l+1}^m}{h^2}$$ (32)

$$u_l^{m+1} = Dk\frac{u_{l-1}^m-2u_l^m+u_{l+1}^m}{h^2}+u_l^m$$

$$u_l^{m+1} = (1-2r)u_l^m+r(u_{l-1}^m+u_{l+1}^m), \quad\mbox{with }r=\frac{Dk}{h^2}.$$ (33)

with $k$ being the step size in time direction and $h$ being the step size in space direction. Crucial to the convergence of this method is the condition

$$r=\frac{k}{h^2}\le\frac{1}{2}.$$

With this approximation we obtain the value of $u(x,t)$ at position $x$ at time $t$ ($u_l^m$) from the values of $x$ at time $t-1$ and the neighbours of $x$ ($x-1$, $x+1$) at time $t-1$.