Variational design of finite-difference schemes for initial value problems with an integral invariant

Abstract

A technique for deriving a finite-difference scheme to solve initial value partial-differential equations is presented. The set of partial-differential equations is assumed to possess one or more invariant integral quantities. In fluid dynamics, the integral of the total energy over the domain is frequently assumed invariant. This technique is based on the variational method and constrains the finite-difference scheme to satisfy the conservation law(s). The technique is discussed by considering, as an example, a set of linear, shallow-water equations on a rotating plane, and extending it to a nonlinear case.