Numerical method for nonlinear wave equation

Question

I need to solve the following nonlinear wave equation numerically

$U_{tt}=(1+\epsilon U_{x}^2)U_{xx}$

with Initial conditions.

what is the best method for solving it? I tried the finite elements method but I did not get a reasonable result?

Thanks

Answer 1

I would convert that into a system of two equations. Let $v = U_t$ and $w = -U_x$. Then you have a following pair of equations $$ v_t + (1 + \epsilon w^2) w_x = 0\\ w_t + v_x = 0 $$ These equations can be rewritten in conservative form as $$ \frac{\partial \mathbf Z}{\partial t} + \frac{\partial \mathbf F(\mathbf Z)}{\partial x} = 0 $$ with $\mathbf Z = (v, w)^\top$ and $$ \mathbf F(\mathbf Z) = \begin{pmatrix} w + \frac{\epsilon}{3} w^3\\ v \end{pmatrix}. $$ Now apply any method for a nonlinear hyperbolic system - Courant-Isaacson-Rees, Lax-Wendroff, or more complicated like some TVD, MUSCL, ENO/WENO or discontinious Galerkin scheme.

PS. It is not clear for me whether your problem may have discontinious solutions. But if it does, you may want to search for a correct conservative form, since the one that I've provided may violate the actual Hugoniot conditions, which can be obtained only from integral formulation of the conservative laws, not the PDE itself.