# 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. – David Ketcheson 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. – David Ketcheson 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. – David Ketcheson Sep 6 '15 at 13:48

## 1 Answer

Due to the potential presence of elliptic regions in your solution, it's not clear to me if equation you are considering has a well-posed initial value problem (IVP). In general for mixed-type PDE one does not have a well-posed IVP; one typically needs to impose other types of boundary conditions. If you are able to find boundary conditions for the boundary of some region in the t-x plane, you may be able to use a numerical procedure outlined in sec. IV of

In that section, the authors numerically solve and equation of the form (their Eq. 33)

$$0 = (A - \partial_x\phi) \partial_x^2\phi + \partial_y^2\phi$$

which is elliptic when $$\partial_x\phi and hyperbolic when $$\partial_x\phi > A$$.

If you are looking for more work on numerical solutions to mixed-type pde, hyperbolic-elliptic type pde seem to appear somewhat frequently in, for example, aeronautics. Unfortunately the specific method one uses depends a lot on the PDE in question.