In this presentation we discuss various continuum models, i.e., based on partial differential equations, and their computational realizations. As frequently is the case, some (scalar) submodels are amenable to analysis while their coupled complex counterparts may not be.
We begin with an overview of challenges in analysis and approximation of scalar conservation laws. The well-known difficulties include the development of singularities from smooth initial data and the instabilities in numerical discretization schemes of high order. Then we discuss conservation laws with additional monotone memory terms which (may) lead to an increased smootheness of solutions. The extended models describe the phenomena of non-equilibrium, double-porosity, and hysteresis occuring, e.g., in preferential adsorption of methane and carbon dioxide in coalbeds. We present our very recent results on convergence of the approximation schemes for these models, but their further extensions to systems of such equations are not readily available. Next, we briefly describe our work on models of methane hydrate evolution; here, the complexity of the underlying multicomponent and multiphase models does not allow comprehensive analysis of well-posedness, or of the regularity of solutions, but we identify mathematical tools developed for problems with inequality constraints to guide the construction of efficient algorithms.
We close by motivating the need to go beyond traditional continuum models and towards hybrid models involving porescale computing, statistical mechanics, and stochastic modeling.