Dual-Family Viscous Shock Waves in n Conservation Laws with Application to Multi-Phase Flow in Porous Media

Abstract

We consider shock waves satisfying the viscous profile criterion in general systems of n conservation laws. We study Si, j dual-family shock waves, which are associated with a pair of characteristic families i and j. We explicitly introduce defining equations relating states and speeds of Si, j shocks, which include the Rankine–Hugoniot conditions and additional equations resulting from the viscous profile requirement. We then develop a constructive method for finding the general local solution of the defining equations for such shocks and derive formulae for the sensitivity analysis of Si, j shocks under change of problem parameters. All possible structures of solutions to the Riemann problems containing Si, j shocks and classical waves are described. As a physical application, all types of Si, j shocks with i>j are detected and studied in a family of models for multi-phase flow in porous media.