I'm trying to solve a complicated PDE for a function $h(x, y,t)$ with mixed derivatives and also a term involving $(h_y)^2$.
Just to try and get my basics right first, suppose we consider ( for $h(y,t)$ ) in a terminal value problem ( means we're solving backwards in time ),
$$ h_t = (h_y)^2 $$
For explicit Euler method we have ( backward differences in time and central differences in space ), $$ \frac{h_n^m-h_n^{m-1}}{\Delta t} = \frac{(h_{n+1}^m)^2-2h_{n+1}^mh_{n-1}^m+(h_{n-1}^m)^2}{4\Delta y^2}\\ h_n^{m-1}=h_n^m - \frac{\Delta t}{4\Delta y^2}[(h_{n+1}^m)^2-2h_{n+1}^mh_{n-1}^m+(h_{n-1}^m)^2] $$ For this do I use something like von Neumann analysis?
For implicit Euler method we have ( forward differences in time and central differences in space ), $$ \frac{h_n^{m+1}-h_n^{m}}{\Delta t} = \frac{(h_{n+1}^m)^2-2h_{n+1}^mh_{n-1}^m+(h_{n-1}^m)^2}{4\Delta y^2}\\ h_n^{m+1}=h_n^m + \frac{\Delta t}{4\Delta y^2}[(h_{n+1}^m)^2-2h_{n+1}^mh_{n-1}^m+(h_{n-1}^m)^2] $$ In this case we need to solve a system of nonlinear equations using Newton Rhapson or something like that.
The actual PDE i'm trying to solve is something more complicated which looks like this,
$$ h_t + \left(\kappa(\theta-y)-\rho\mu\xi\right)h_y + \frac{1}{2}\xi^2yh_{yy} + \frac{1}{2}\gamma(1-\rho^2)\xi^2y h_y^2 + \xi^2y\frac{h_yf_y}{f} \\+ \frac{1}{2}yS^2h_{SS} + \rho\xi y S h_{Sy} =0\\h(S,y,T)= g(S,y) $$ where $\kappa, \theta,\xi, \gamma,\rho$ are constants, and $f$ is another given function.
I would very much appreciate if someone help with pointing me to good references/papers...
Thanks!