A Study of Mass Conserving Function Solution of Differential Equation

An Adaptive Solution Approach using Moving Mesh Technique

Abstract

In this study we consider the adaptive solution of time-dependent partial differential equations using a moving mesh technique. A moving mesh method is developed utilising monitor functions to drive the mesh motion and is based on equidistribution ideas. The method is derived in multidimensions from a conservation of monitor function principle and from this initial principle a conservation law for the monitor function is derived which generates the velocity of the mesh. In dimensions higher than one the mesh velocity is underdetermined for this conservation law, therefore the curl of the velocity field is prescribed to obtain a unique velocity field. The conservation law together with the curl condition is combined to produce an elliptic equation for a mesh velocity potential, from which the mesh velocity can be obtained. The moving mesh equations are solved for the mesh velocity using standard linear finite elements and then a new mesh is constructed by integrating the mesh velocity forward in time using finite differences. The moving mesh method is applied to the adaptive solution of parabolic and hyperbolic equations. The Partial Differential Equations (PDE) with a moving boundary problem is solved taking advantage of scale invariant properties. For this problem the solution to the PDE is obtained through the conservation of monitor function for two different monitor functions a mass and a gradient monitor function.