Error Analysis of an Adaptive Implicit Scheme for Hyperbolic Conservation Laws

Abstract

Implicit time integration of mass conservation equations is necessary to maintain a practically feasible time step for problems with substantial variation in the local CFL number. The adaptive implicit method (AIM) is a mixed implicit-explicit scheme that lowers the computational cost by adaptively applying, in both space and time, implicit time integration to a small number of cells with large local CFL values. The resulting computational efficiency is utilized to improve the accuracy of the solution by employing high-order spatial discretization for both types of cells. We show that regardless of the order of accuracy, the AIM discretization is inconsistent, and that it introduces substantial numerical errors at the interfaces between explicit and implicit cells. We perform an error analysis of discretizations based on AIM to determine the error characteristics with respect to the order of accuracy, as well as the number and propagation rate of the implicit-explicit boundaries. We demonstrate that the linear AIM operators are monotone, nonoscillatory, and stable. We also show that the associated error normalized by $\Delta t$ is bounded for finite, stationary AIM boundaries but increases as a power law when the AIM boundaries move with the speed of the error pulse. We derive the conditions for which large errors and nonmonotonic solutions are obtained. A consistent version of AIM is proposed and compared with the standard AIM scheme. The error analysis and new AIM framework provide a basis for the design and implementation of AIM schemes for problems of practical interest.