Fully Adaptive Solutions of One-Dimensional Mixed Initial-Boundary Value Problems with Applications to Unstable Problems in Combustion

We develop a fully adaptive method for solving one-dimensional mixed initial-boundary value problems. The method adaptively adjusts the number of grid points needed to equidistribute a positive weight function over a given mesh interval at each time level. In addition, when the solution can be described accurately by a fixed number of points, the mesh is moved by extrapolating the nodal positions from earlier time levels. We monitor the solution from one time level to another to insure that the local error per unit step associated with the time differencing method is below some specified tolerance. The method is applied to several unstable problems in combustion.