Convergence analysis of the Hopmoc method

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.

[1]  Jim Douglas,et al.  Numerical solution of two‐dimensional heat‐flow problems , 1955 .

[2]  Willem Hundsdorfer,et al.  Linear stability of the hopscotch scheme , 1989 .

[3]  Jr. Jim Douglas.  Alternating direction iteration for mildly nonlinear elliptic difference equations , 1961 .

[4]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[5]  Paul Gordon,et al.  Nonsymmetric Difference Equations , 1965 .

[6]  J. J. Douglas.  Alternating direction methods for three space variables , 1962 .

[7]  Robert Todd Gregory,et al.  A collection of matrices for testing computational algorithms , 1969 .

[8]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[9]  J. Douglas,et al.  A general formulation of alternating direction methods , 1964 .

[10]  A. R. Gourlay,et al.  Hopscotch: a Fast Second-order Partial Differential Equation Solver , 1970 .

[11]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[12]  J. J. Douglas.  On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods , 1955 .

[13]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[14]  Dean G. Duffy.  Unsteady viscous flow around a circular cylinder found by the hopscotch scheme on a vector processing machine , 1990 .