Accurate Local Time Stepping Algorithms for Calculating Unsteady and Steady Solutions
by
Jianping Zhu
Mississippi State University
Coauthors: Kiran Kumar V. Adhikarala (Mississippi State University), Edward Luke (Mississippi State University), Pasquale Cinnella (Mississippi State University)

Local time stepping algorithms have been traditionally used to improve computational efficiency in the calculation of steady solutions for CFD problems. The basic idea is to advance solutions in the temporal dimension using different time steps at different spatial grid points. The typical practice is to use a uniform CFL number for all grid points, which effectively advances solutions using a large time step for the spatial points with a large grid spacing, and small time steps for the spatial area where grid points are more densely distributed. Since time consistency is not maintained in the traditional local time stepping algorithms, they can not be used to calculate accurate unsteady solutions.

Our previous work focused on the development of accurate explicit local time stepping algorithms for calculating unsteady solutions. In this presentation, we will discuss extensions to implicit local time stepping algorithms. The basic idea is to use proper interpolation and extrapolations to match solutions calculated using different time steps, and to advance solutions at all spatial grid points to the same time level, which means that multiple time steps have to be used for the solutions calculated using small time steps. It will be shown that for systems of first order hyperbolic equations, an interpolation or extrapolation of order p-1 is needed in order to maintain consistency and accuracy of a time integration algorithm of order p. Numerical results on simple model equations will also be discussed in the presentation.

Date received: January 14, 2000