Numerical methods for a second order PDE boundary value problem Is there any numerical method (like the method of splitting of variables for the equation of 2D-diffusion) for solving the following boundary value problem
$$\left\{\begin{aligned}&\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^2 u}{\partial x^2};\\
 &(x,y,t)\in\Omega=[0,a]\times[0,b];\\
&u(x,y,0)=u_{0}(x,y); \\
&u(\partial\Omega)=0? \end{aligned}\right.$$
 A: Here is one suggestion. 
First notice your equation is:
$$
\frac{\partial u}{\partial t} = \nabla \cdot (A\nabla u),
$$
where
$$
A  = \begin{pmatrix}1& 0 \\
1 &0\end{pmatrix}.
$$
$A$ is not positive definite here, the right side is not an elliptic operator, traditional numerical methods like finite difference or finite element is good for positive definite $A$.
Now you can make a little perturbation on $A$, say 
$$
A_{\epsilon}  = \begin{pmatrix}1& 0 \\
1 &\epsilon\end{pmatrix},
$$
for $\epsilon >0$ but small, now the eigenvalues of $A$ are $1$ and $\epsilon$, which gives you a well-posed diffusion equation with diffusion matrix $A_{\epsilon}$:
$$
\frac{\partial u_{\epsilon}}{\partial t} = \nabla \cdot (A_{\epsilon}\nabla u_{\epsilon}).
$$
Use any of your favorite numerical methods to solve above equation and let $\epsilon \to 0$ see what happened on $u_{\epsilon}$.

About finite differences:
$$
\frac{\partial^2 u}{\partial x^2}\Bigg|_{(i,j);t=t_n} \approx \frac{u^n(i+1,j) -2u^n({i,j})+ u^n({i-1,j})}{(\Delta x)^2} \\
=\frac{1}{\Delta x}\left(\frac{u^n(i+1,j) -u^n(i,j)}{\Delta x}-\frac{u^n(i,j)-u^n({i-1,j})}{\Delta x}\right),
$$
this is how you approximate the second partial derivative in $x$. As you can see, this is nothing but approximating $\partial/\partial x$twice. Mimicing above you could use the following to approximate the mixed partial derivative
$$
\frac{\partial   }{\partial y}\left(\frac{\partial  u}{\partial x}\right)\Bigg|_{(i,j);t=t_n} \approx \frac{1}{2\Delta y} \left( \frac{\partial u}{\partial x} \Bigg|_{(i,j+1);t=t_n}  -  \frac{\partial u}{\partial x} \Bigg|_{(i,j-1);t=t_n}\right)
\\
\approx \frac{1}{2\Delta y}\left(\frac{u^n(i+1,j+1) -u^n(i,j+1)}{\Delta x}-\frac{u^n(i,j- 1)-u^n({i-1,j- 1})}{\Delta x}\right),
$$
if you use the central difference to approximate the partial derivative in $y$.
