An h-r Moving Mesh Method for One-Dimensional Time-Dependent PDEs

We propose a new moving mesh method suitable for solving timedependent partial differential equations (PDEs) in R which have fine scale solution structures that develop or dissipate. A key feature of the method is its ability to add or remove mesh nodes in a smooth manner and that it is consistent with r-refinement schemes. Central to our approach is an implicit representation of a Lagrangian mesh as iso-contours of a level set function. The implicitly represented mesh is evolved by updating the underlying level set function using a derived level set moving mesh partial differential equation (LMPDE). The discretized LMPDE evolves the level set function in a manner that quasi-equidistributes a specified monitor function. Beneficial attributes of the method are that this construction guarantees that the mesh does not tangle, and that connectivity is retained. Numerical examples are provided to demonstrate the effectiveness of our approach.