Efficient and Accurate Local Time Stepping Algorithms for Multi-Rate Problems

A multi-rate problem is described by a system of coupled partial differential equations with different time scales associated with different equations in the system. The numerical solutions to such systems are usually calculated using a time step determined by the most restrictive time scale in the system for stability and accuracy considerations. We demonstrate in this paper that this time step could be excessively small and unnecessary in many situations, and discuss a more efficient time integration method that uses different time steps for different equations depending on their time scales. Numerical results will be presented to demonstrate significant improvement in computational efficiency.