the adaptive solution of evolutionary partial differential equations (PDEs) in one space dimension. For unsteady A coordinate transformation approach is described that enables pseudospectral methods to be applied efficiently to unsteady differproblems, adaptive methods may be classified as static or ential problems with steep solutions. The work is an extension of a dynamic. In the static approach the numerical solution is method presented by Mulholland, Huang, and Sloan for the adaptive advanced in time on a fixed nonuniform grid, and after pseudospectral solution of steady problems. A coarse grid is genereach step (or series of steps) a regridding is carried out to ated by a moving mesh finite difference method that is based on give new nodal locations that are adapted in some sense to equidistribution, and this grid is used to construct a time-dependent coordinate transformation. A sequence of spatial transformations the computed solution. This is followed by an interpolation may be generated at discrete points in time, or a single transformaprocess that yields approximations on the new grid to serve tion may be generated as a continuous function of space and time. as initial values for the next time step. In the static method
The differential problem is transformed by the coordinate transforthere is no essential coupling between the discretisation mation and then solved using a method that combines pseudospectral discretisation in space with a suitable integrator in time. Numeri-of the PDE and the grid generation. The reader is referred cal results are presented for unsteady problems in one space to papers by Sanz-Serna and Christie [16] and Bieterman dimension. แฎ 1997 Academic Press and Babus หka [2] for illustrations of the static regridding approach.
In the dynamic adaptive methods there is a coupling This function is usually some measure of the local computa-Huang, Ren, and Russell [8,9] have presented several tional error or the local solution variation. The text by moving mesh PDEs that are based on the concept of equi-Thompson et al. [17] and the paper by Huang and Sloan distribution. Their moving mesh equations have solutions [11] give an interpretation of equidistribution in the context that are continuous in space and time, and the equations of adaptive grid generation for steady, one-and two-diare designed so that the nodal locations satisfy an equidismensional problems. An equidistribution principle is detribution principle (EP). In practice, the moving mesh veloped in [11] and it is used to formulate a finite difference equations in [8,9] are conjoined with the given PDE in grid generation algorithm in two space dimensions.
The aim of this paper is to describe a robust method for the discretisation process.
Our purpose here is to present a moving mesh method that incorporates a pseudospectral (PS) post-processing * Supported by the Engineering and Physical Sciences Research Counstep. We give an extension of a method for the adaptive cil (EPSRC).