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

Highlights

• Numerical scheme which combines two popular methods (ELLAM and MMOC).

• Combination such that the benefits are preserved, whilst drawbacks are eliminated.

• A new and efficient algorithm to achieve local volume conservation.

• Convergence theorem via the generic framework of Gradient Discretisation Method.

Abstract

We propose a combination of the Eulerian Lagrangian Localised Adjoint Method (ELLAM) and the Modified Method of Characteristics (MMOC) for time-dependent advection-dominated 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 (Cheng et al., 2018), is provided which can be extended to obtain the convergence of GEM scheme.