Convergence analysis of the Hopmoc method

Abstract

The Hopmoc method combines concepts of the modified method of characteristics (MMOC) and the Hopscotch method. First, Hopmoc resembles Hopscotch because it decomposes the set of grid points into two subsets. Namely, both subsets have their unknowns separately updated within one semi-step. Furthermore, each subset undergoes one explicit and one implicit update of its unknowns in order to lead to a symmetrical procedure. Such decomposition inspired the use of a convergence analysis similar to the one used in alternating direction implicit methods. Secondly, the steps are evaluated along characteristic lines in a semi-Lagrangian approach similar to the MMOC. In this work, both consistency and stability analysis are discussed for Hopmoc applied to a convection–diffusion equation. The analysis produces sufficient conditions for the consistency analysis and proves that the Hopmoc method presents unconditional stability. In addition, numerical results confirm the conducted convergence analysis.