An explicit, adaptive grid algorithm for one-dimensional initial value problems Original Research Article

A new explicit, adaptive grid algorithm for use in the solution of one-dimensional initial value problems is described. The method is based on the concept of tracking individual features of the physical solution. Properties of these real features are used to construct an artificial resolution function corresponding to a finite number of artificial features, each of which is then used to track a real feature. The artificial resolution function is constructed in such a way that it is analytically integrable, which allows a very stable and simple grid generation. Moving the artificial features is much more straightforward than trying to move the grid-points directly. In particular, features which interact, or even move off the grid can be treated adequately, without adversely affecting the monotonicity and smoothness of the grid. The results of several experiments are presented. These include a number of problems using test artificials, the aim of which is to mimic many of the situations involving sharp discontinuities which arise in the solution of hyperbolic PDEs. In addition, the algorithm is implemented in conjunction with a hydrodynamical test problem.