The modified method of characteristics with adjusted advection

Abstract

The MMOC procedure for approximating the solutions of transport-dominated diffusion problems does not automatically preserve integral conservation laws, leading to (mass) balance errors in many kinds of flow problems. The variant, called the MMOCAA, discussed herein preserves the conservation law at a minor additional computational cost. It is shown that its solution, in either Galerkin or finite difference form, converges at the same rates as were proved earlier by Douglas and Russell for the standard MMOC procedure. Mathematics Subject Classification (1991): 65M25

A Highly Accurate Modified Method of Characteristics for Convection–dominated Flow Problems

“…The conventional MMC methods have been widely used to approximate the solution of (4.1), compare [4,9,8,16,24,26,30,29] for analysis and computational aspects of these methods for general transport-diffusion problems. As applied to (4.1), the MMC consists of an advective-diffusive splitting of the equations along the characteristics.…”

“…In addition, it has been observed that the error of the conventional MMC in solving advectiondiffusion problems decreases as the time step increases in a certain range of parameters. Analysis of the convergence and stability of the conventional MMC has been carried out in many papers, among others, in [4,8,9,24,33]. For convection-dominated equations with a diffusion bounded away from zero, the authors of [9] show an optimal error estimate…”

A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations

“…Nevertheless, the derived error estimate suffers from the same limitation as the earlier error estimates for many other characteristic methods, e.g. the modified method of characteristics (MMOC) [6], the modified method of characteristics with adjusted advection (MMOCAA) [7], and the Eulerian-Lagrangian localized adjoint method (ELLAM) for unsteady-state advection-diffusion equations with general boundary conditions [8]. Namely, the generic constant in the estimates may depend inversely on the vanishing diffusion parameter.…”

A Mass-Conservative Characteristic Finite Element Scheme for Convection-Diffusion Problems

“…Among them the procedure of the characteristic method is natural from the physical point of view since it approximates particle movements, and it is attractive from the mathematical point of view since it symmetrizes the problem. Many authors have contributed to develop, analyse and apply characteristic finite element schemes; see [1], [4], [6], [7], [8], [13], [14], [15], [16], [21] and references therein.…”

“…In the framework of characteristic methods it it not trivial to maintain this property. Some schemes have been proposed and studied from this point [1], [6], [8], [21].…”