Moving Mesh Techniques Based upon Equidistribution, and Their Stability

Various aspects of the moving mesh problem are investigated for the solution of partial differential equations (PDEs) in one space dimension. In particular, methods based (explicitly or implicitly) upon an equidistribution principle are studied. It is shown that equidistribution implicitly corresponds to finding a solution to a PDE involving a new set of computational coordinates. Implementation of a discrete version of equidistribution to compute a moving mesh corresponds to solving a weak form of the PDE. The stability of equidistribution is discussed, and it is argued that stability can be significantly affected by the way in which this solution process is carried out. Simple moving mesh methods are constructed using this framework, and numerical examples are given to illustrate their robustness.