A Self-adaptive Time Integration Algorithm for Solving Partial Differential Equations

Non-uniform spatial grids are commonly used to resolve locally fast-changing physical phenomena in space. If a traditional explicit time integration scheme is used to advance solutions in the temporal dimension, the step size is restricted by the stability criterion, which is in turn dictated by the smallest grid spacing in the spatial dimensions. It turns out that the excessively small time step enforced by the smallest spatial grid spacing in a domain is usually unnecessary for most of the grid points with larger grid spacing. A new adaptive time integration method is introduced in this paper to improve the computational efficiency. The basic idea is to use different time step sizes at different spatial grid points. The stability criterion is still satisfied at all grid points by using different time step sizes. In this way, the time step size adjusts automatically based on the local spatial grid spacing. Complexity analysis and implementation details are also discussed in the paper. The non-linear Burger's equation is used in the numerical experiment. Both the complexity analysis and numerical computations demostrate significant improvement of computational efficiency.