What type of numercial methods are there to solve PDE of the sorts of: $$f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))$$ $$u(x,0)=G_1(x) , \frac{\partial u(x,0)}{\partial t}=H_1(x) ,u(0,t)=G_2(t) , \frac{\partial u(0,t)}{\partial x} = H_2(t)$$ 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 conetents, 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 posted this question also here: https://scicomp.stackexchange.com/questions/20643/numerical-methods-for-solving-a-hyperbolic-nonlinear-pde

hope it doesn't bother anyone.