Integral Preserving Numerical Methods on Moving Grids

Abstract

Integral preservation for ordinary and partial differential equations is defined, and the integral preserving discrete gradient methods and discrete variational derivative methods on fixed grids are given, with a formal definiton for the latter in a general grid case.General moving grid methods are presented, with special emphasis on grid movement strategies. The integral preserving modified discrete variational derivative methods for moving grids are introduced, and integral preserving projection methods are shown to be a subset of these. New interpolation techniques are introduced, and a general solution procedure for implementing the methods is presented. Two alternative integral preserving moving grids methods are briefly presented.The modified discrete variational derivative methods are applied on two different partial differential equations, and numerical experiments are performed on moving grids. The integral preserving property of the methods is demonstrated and their advantages and applications are discussed.