On Operator Splitting for the Viscous Burgers' and the Korteweg-de Vries Equations

Abstract

We discuss numerical quadratures in one and two dimensions, which is followed by a discussion regarding the differentiation of general operators in Banach spaces. In addition, we discuss the standard and fractional Sobolev spaces, and prove several properties for these spaces.We show that the operator splitting methods of the Godunov type and Strang type applied to the viscous Burgers equation, and the Korteweg-de Vries (KdV) equation (and other equations), have the correct convergence in the Sobolev spaces. In the proofs we use the new framework originally introduced in [11].We investigate the Godunov method and Strang method numerically for the viscous Burgers equation and the KdV equation, and present different numerical methods for the subequations from the splitting. We numerically check the convergence rates for the split step size, in addition with other aspects for the numerical methods. We find that the operator splitting methods work well numerically for the two equations. For the viscous Burgers equation, we find that several combination of numerical solvers for the subequations work well on the test problems, while we for the KdV equation find only one combination of numerical solvers which works well on all test problems.