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How to solve the third order time dependent partial differential equation

$$u_t + 6u_x + u_xxx = 0$$

in weak form using galerkin finite element method?

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  • $\begingroup$ *correction = should be finite element method $\endgroup$
    – Hooi Mun Hoe
    Commented Aug 5, 2018 at 5:02

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Although I haven't worked on 3rd order PDEs but I believe, because of odd orders in space, we cannot use the normal Ritz-Galerkin method where we choose both Trial and Test functions from the same space. We will have to go to the Petrov-Galerkin methods which are suited for this. Especially, handling the convective term requires Stabilization (like Upwinding, SUPG, Least squares etc.). And all of these can be done in Method of lines way, which is you write the space terms in weak form add stabilization, form equations for each node by testing with the test functions and then do the time marching for all of the nodes.

Additionally, it being third order you might like to have atleast twice differentiable in weak sense, and constructing twice differentiable trial or test functions is hard task (you can't start with linear elements which are only continuous). So it might be better to use Discontinuous Galerkin approach clubbed with Petrov-Galerkin idea adding stabilization!

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