Numerical methods for solving a mixed type nonlinear PDE

What type of numerical methods are there to solve PDE of the sorts of: \begin{align} &f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))\\ &u(x,0)=G_1(x)\\ &\frac{\partial u(x,0)}{\partial t}=H_1(x)\\ &u(0,t)=G_2(t) , \frac{\partial u(0,t)}{\partial x} = H_2(t) \end{align} Where $f,\ g,\ F,\ u \in C^\infty(x,t),\ G_i,\ H_i \in C^\infty$.

Specifically I had in mind the PDE: $$u_{xx}u^3-\sin(xt)u_{tt} = u$$

But the general PDE is as above; I looked at Polyanin's second edition Handbook of Nonlinear PDE table of contents, and didn't find something similar, obviously I look at numerical solutions since an analytical solution doesn't seem plausible, but if there is I wouldn't mind. :-)

I also posted this question in MathOverflow, where one answer pointed out that my problem is likely not well-posed.

• Are you sure your example is hyperbolic? At t=0 or x=0, it becomes elliptic, and depending on the domain it will be parabolic in some regions. Before looking for numerical solutions I would wonder if it is well-posed. Sep 5 '15 at 11:26
• @DavidKetcheson can't we have problems where the PDE changes from being elliptic to being parabolic and hyperbolic in different regions?
– Alan
Sep 5 '15 at 12:03
• Yes, we can and do. But we don't simply call them "hyperbolic" in that case. The title of the question seems wrong. And the numerical methods required in the different regions will be different. Sep 6 '15 at 6:09
• @DavidKetcheson I changed the title of my post. please do suggest me some numerical methods to use here?
– Alan
Sep 6 '15 at 6:55
• Sorry, but I think it makes no sense to recommend a numerical method in light of the well-posedness issue. Sep 6 '15 at 13:48

$$0 = (A - \partial_x\phi) \partial_x^2\phi + \partial_y^2\phi$$
which is elliptic when $$\partial_x\phi and hyperbolic when $$\partial_x\phi > A$$.