Local Time-stepping Methods Research Articles

Adaptive techniques for time-dependent problems

Both a priori and a posteriori error estimation techniques have played an important role in the adaptivity of numerical methods for static problems with important localized phenomena. Timedependent problems with localized temporal effects offer an even greater challenge to effective adaptivity. The need for flexible data structures and solution algorithms to accommodate moving patch refinement to follow moving sharp fronts in fluid flow applications is discussed. In parabolic problems, often transient temporal effects are generated by spatially fixed but dynamic source terms. Local time-stepping methods are presented to treat these transient phenomena. Stability and accuracy of the methods are considered. Results of numerical experiments are presented that indicate the relative roles of the local temporal and spatial discretizations in the performance of the methods.