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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.

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  • $\begingroup$ Another site of relevance for this question is scicomp.SE. $\endgroup$ – Igor Khavkine Sep 4 '15 at 9:55
  • $\begingroup$ Crossposted there: scicomp.stackexchange.comquestions/20643. (Please don't crosspost, at least not without pointing it out! Otherwise people waste their time writing an answer you have already received at another site.) $\endgroup$ – Christian Clason Sep 4 '15 at 10:46
  • $\begingroup$ @ChristianClason I added a remark in both posts clearing out that I posted this question in two different stackexchange sites. $\endgroup$ – Alan Sep 4 '15 at 10:56
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I will let the SciComp people address the issue of the actual numerical method. Instead I will point out that your PDE, as written, is not necessarily hyperbolic. In particular, you see that $\sin(xt)$ can change signs, and $u^3$ can change signs. This means that depending on the parameters your PDE is actually of mixed type: at some regions it is hyperbolic, on some regions it is elliptic, and on some boundary portions it is degenerate/parabolic.

Unless you can a priori rule out these type changes (which I don't see how, since $\sin(xt)$ is independent of the solution $u$ or its initial data), if you try to solve this problem numerically as an initial value problem, your code will have "terrible, horrible, no good very bad" instabilities.

Also, when $t = 0$ your equation reduces to an ODE on the line. So your initial data $G_1$ is actually not free. Because all these considerations, it is not even clear to me that your problem is well-posed, never mind admitting a numerical solution method.

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  • $\begingroup$ In fact, I'd be surprised if it were well-posed since the left boundary conditions are over-specified -- that's a lateral Cauchy problem, which for linear elliptic equations is literally the textbook example of an ill-posed problem. (Of course, with a nonlinear equation like this, who knows...) $\endgroup$ – Christian Clason Sep 4 '15 at 14:34
  • $\begingroup$ Hi Willie Wong, can you prove that the problem as stated is ill-posed? What if I were to change $\sin(xt)$ with $\sin^2(u)$, and $u^3$ with $u^2$.BTW, I have another question from an ultrahyperbolic PDE which I hope you can address it appears here:mathoverflow.net/questions/212978/… $\endgroup$ – Alan Sep 4 '15 at 14:58
  • $\begingroup$ @Alan: if you make the replacement as you indicated, then away from the points where $\{u(x,t) = k\pi\}$ your equation is hyperbolic. That gives a bit more hope. $\endgroup$ – Willie Wong Sep 4 '15 at 20:12
  • $\begingroup$ BTW, Willie why does it matter for well posedness that the problem is the same type in most parts of the domain? (the types are elliptic , hyperbolic and parabolic). $\endgroup$ – Alan Sep 5 '15 at 12:07
  • $\begingroup$ How do you suggest me to show ill posededness of the original problem, in case I pick as initial conditions trig functions such as sine and cosine? $\endgroup$ – Alan Sep 6 '15 at 16:39

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