A stabilized Galerkin method for a third-order evolutionary problem

Abstract:

The periodic initial value problem for the partial differential equation ${u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac {\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}} - \delta {u_{tx}} = 0$, $\varepsilon$, $\delta > 0$, arises in fluidization models. The numerical integration of the problem is a difficult task in that many "reasonable" finite difference and finite element methods give rise to unstable discretizations. We show how to modify the standard Galerkin technique in order to stabilize it. Optimal-order error estimates are derived and the results of numerical experiments are presented. The stabilization technique suggested in the paper can be interpreted as rewriting the problem in Sobolev form and would also be useful for other equations involving terms of the form ${u_t} - \delta {u_{tx}}$.