Tensor-product adaptive grids based on coordinate transformations

In this paper we discuss a two-dimensional adaptive grid method that is based on a tensor-product approach. Adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations (PDEs) and ordinary differential equations. A traditional and widely used form of adaptivity is the concept of equidistribution, which is well-defined and well-understood in one space dimension. The extension of the equidistribution principle to two or three space dimensions, however, is far from trivial and has been the subject of investigation of many researchers during the last decade. Besides the nonsingularity of the transformation that defines the nonuniform adaptive grid, the smoothness of the grid (or transformation) plays an important role as well. We will analyse these properties and illustrate their importance with numerical experiments for a set of time-dependent PDE models with steep moving pulses, fronts, and boundary layers.