I am atempting to discretize the following two PDEs:
$$\ \frac\partial{\partial x}(T\frac{\partial h}{\partial x})=-N+S\frac{\partial h}{\partial t}-\frac{\partial}{\partial x}(\tau\frac{\partial \zeta}{\partial x})$$
$$\ \frac\partial{\partial x}(\tau\frac{\partial \zeta}{\partial x})=n\frac{\partial \zeta}{\partial t}-\frac{\partial}{\partial x}(\frac\tau v \frac{\partial h}{\partial x})$$
Where $T,\tau,v,N,S$ are constants and do not depend on $x$ or $t$.
My intention is to produce marching equations which I can use to step through time. i.e. find the values of $h$ and $\zeta$ at time $t+k\Delta t$. All inputs would be discrete points.
I tried using finite difference methods and got the following two equations: $$h_i^{k+1}=C_Th_{i+1}^k+(1-2C_T)h_{i}^k+C_Th_{i-1}^k+\frac{\Delta tN}{S}+C_{\tau}\zeta_{i+1}^k-2C_{\tau}\zeta_{i}^k+C_{\tau}\zeta_{i-1}^k$$ $$\zeta_i^{k+1}=D_{\tau}\zeta_{i+1}^k+(1-2D_{\tau})\zeta_{i}^k+D_{\tau}\zeta_{i-1}^k+\frac{D_{\tau}}{v}\zeta_{i+1}^k+\frac{D_{\tau}}{v}\zeta_{i}^k+\frac{D_{\tau}}{v}\zeta_{i-1}^k$$
Where $C_T=\frac{\Delta tT}{S\Delta x^2}, C_{\tau}=\frac{\Delta t\tau}{S\Delta x^2}, D_{\tau}=\frac{\Delta t\tau}{n\Delta x^2}$
Does this look correct?