Efficient strategies for solving the variable Poisson equation with large contrasts in the coefficients

Abstract

Discrete versions of Poisson’s equation with large contrasts in the coefficients result in very ill-conditioned systems. Thus, its iterative solution represents a major challenge, for instance, in porous media and multiphase flow simulations, where considerable permeability and density ratios are usually found. The existing strategies trying to remedy this are highly dependent on whether the coefficient matrix remains constant at each time iteration or not. In this regard, incompressible multiphase flows with high-density ratios are particularly demanding as their resulting Poisson equation varies along with the density field, making the reconstruction of complex preconditioners impractical. This work presents a strategy for solving such versions of the variable Poisson equation. Roughly, we first make it constant through an adequate approximation. Then, we block-diagonalise it through an inexpensive change of basis that takes advantage of mesh reflection symmetries, which are common in multiphase flows. Finally, we solve the resulting set of fully decoupled subsystems with virtually any solver. The numerical experiments conducted on a multiphase flow simulation prove the benefits of such an approach, resulting in up to 6.6x faster convergences.

Adel Alsalti-Baldellou, Xavier Àlvarez-Farré, F. Xavier Trias and Assensi Oliva have been ´ financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Adel ` Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019- DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively. Andrey Gorobets has been supported by the RSF project 19-11-00299.

Peer Reviewed

Postprint (published version)