This technique allows us to trasform e.g, Heston, Basket pdes to pdes with no mixed derivatives.

Taking Heston[$](x,y)[$] the original pde is

[$] \frac{1}{2} y x^2 \frac{\partial^2 u}{\partial x^2} + \rho\sigma xy\frac{\partial^2 u}{\partial x \partial y} + \frac{1}{2} \sigma^2 y \frac{\partial^2 u}{\partial y^2} [$] ... + ... lower-order terms

By a change of coordinates

[$]\xi = \sigma\rho log(x) - y[$] and [$]\eta = \sigma\sqrt{(1-\rho^2)}log(x)[$]

we get a Heston[$](\xi, \eta)[$] with no mixed derivatives for the principal part of the pde (which can't be bad).

[$] A \frac{\partial^2 u}{\partial \xi^2} + B \frac{\partial^2 u}{\partial \eta^2} [$] ... + ... lower-order terms

where [$]A = B = \frac{1}{2} \sigma^2 y (1 -\rho^2)[$] and [$] y = (\rho/(1-\rho^2)\eta - \xi[$]

The question now is:

"The original domain is the positive quarter plane; what is the [$]\xi-\eta[$] domain and most importantly how to prescribe numerical boundary conditions?

The answer could be easy or difficult.