A review of moving mesh methods for the numerical solution of PDEs

Accurate modelling of scientific problems that are governed by partial differential equations (PDEs) with steep solution regions often involves high computational cost if a uniform mesh is used. In recent years a family of methods---moving mesh methods---has been developed that adapts the mesh to features of the computed solution. The nodal density is high in regions of high solution variation and low in regions where the solution variation is small. The talk describes moving mesh methods that are based on the idea of equidistribution (see, for example, W Huang and R D Russell, SIAM J Sci Comput 20 (1999) 998-1015). These methods utilise a PDE to evolve the mesh in a manner that accurately captures sharp fronts with a relatively small number of mesh points. The complete solution process involves the combined numerical solution of a moving mesh PDE and the governing system of physical PDEs. Numerical results are referenced to demonstrate the effectiveness of the methods.