A Novel Semi-Analytical Multiple Invariants-Preserving Integrator for Conservative PDEs

Abstract

Many conservative partial differential equations such as the Korteweg-de Vries (KdV) equation, the nonlinear Schrödinger equation, and the Klein-Gordon equation have more than one invariant functionals. In this paper, we propose the definition of the discrete variational derivative, based on which, a novel semi-analytical multiple invariants-preserving integrator for the conservative partial differential equations is constructed by projection technique. The proposed integrators, constructed by applying a projection technique to existing numerical methods, are shown to preserve the same order of accuracy as their underlying base integrators. For applications, some concrete mass-momentum-energy-preserving integrators are derived for the KdV equation.