A combined GDM--ELLAM--MMOC scheme for advection dominated PDEs

TL;DR

This paper introduces the GEM scheme, combining ELLAM and MMOC methods to improve mass conservation and computational efficiency in advection-dominated PDEs, supported by theoretical analysis and numerical comparisons.

Contribution

The paper develops a novel combined scheme (GEM) that leverages the strengths of ELLAM and MMOC, with proven mass conservation and convergence properties.

Findings

GEM scheme effectively balances mass conservation and computational ease.

Numerical results show GEM outperforms individual ELLAM and MMOC schemes.

Convergence of GEM scheme is supported by theoretical analysis.

Abstract

We propose a combination of the Eulerian Lagrangian Localised Adjoint Method (ELLAM) and the Modified Method of Characteristics (MMOC) for time-dependent advection-domina\-ted PDEs. The combined scheme, so-called GEM scheme, takes advantages of both ELLAM scheme (mass conservation) and MMOC scheme (easier computations), while at the same time avoids their disadvantages (respectively, harder tracking around the injection regions, and loss of mass). We present a precise analysis of mass conservation properties for these three schemes, and after achieving global mass balance, an adjustment yielding local volume conservation is then proposed. Numerical results for all three schemes are then compared, illustrating the advantages of the GEM scheme. A convergence result of the MMOC scheme, motivated by our previous work on the convergence of ELLAM schemes, is provided, which can be extended to obtain the convergence of GEM scheme.