An adaptive moving grid method for one-dimensional systems of partial differential equations

Abstract

We describe a fully adaptive, moving grid method for solving initial-boundary value problems for systems of one-space dimensional partial differential equations whose solutions exhibit rapid variations in space and time. The method, based on finite-differences, is of the Lagrangian type and has been derived through a co-ordinate transformation which leads to equidistribution in space of the second derivative. Our technique is “intermediate” between static regridding methods, where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled. In our approach, the computation of the moving grids and the solution on these grids are carried out separately, while the nodes are moved at each time-step. Two error monitors have been implemented, one to govern the time-step selection and the other to eventually adapt the number of moving nodes. The method allows the use of different moving grids for different components in the PDE system. Numerical experiments are presented for a set of five sample problems from the literature, including two problems from combustion.