On the approximation of solutions of the generalized Korteweg–de Vries–Burger's equation

We propose numerical schemes for approximating periodic solutions of the generalized Korteweg—de Vries—Burgers equation. These schemes are based on a Galerkin-finite element formulation for the spatial discretization and use implicit Runge—Kutta (IRK) methods for time stepping. Asymptotically optimal rate of convergence estimates can be obtained in terms of the spatial and temporal discretization parameters. In particular, the temporal rates are the classical ones, i.e. no order reduction occurs. We also apply Newton's method, to solve the system of nonlinear equations. Indeed, Newton's method yields iterants that converge quadratically and preserves the optimal rates of convergence.