Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

Abstract

We propose finite-volume schemes for general continuity equations which\npreserve positivity and global bounds that arise from saturation effects in the\nmobility function. In the case of gradient flows, the schemes dissipate the\nfree energy at the fully discrete level. Moreover, these schemes are\ngeneralised to coupled systems of non-linear continuity equations, such as\nmultispecies models in mathematical physics or biology, preserving the bounds\nand the dissipation of the energy whenever applicable. These results are\nillustrated through extensive numerical simulations which explore known\nbehaviours in biology and showcase new phenomena not yet described by the\nliterature.

Topics & Concepts