Background

The background needed to meet the objectives of this work includes entropy viscosity methods, and the TCAT approach for model formulation and the calculation of a physically-based entropy pro- duction. The status of knowledge on these two components are considered in turn.

Standard conforming Galerkin finite elements can produce non-physical oscillations for hyper- bolic or advection-dominated problems that are not strictly hyperbolic. To reduce these oscillations, either artificial viscosity (or diffusion) can be added to the solution or the computational mesh can be refined, which lowers the mesh Peclet number (Baliga and Patankar, 1980; Carver and Hinds, 1978). Both of these options have shortcomings. Adding artificial viscosity to a solution will dampen oscil- lations but will also smear the solution and the amount of artificial viscosity to add is not knowna priori. For large simulations, refining the underlying computational mesh may not be possible given a constraint on available computational resources.

For the artificial viscosity approach, the amount of required viscosity to eliminate oscillations is dependent on the steepness of the solution relative to a given mesh resolution, with more artificial vis- cosity needed near large gradients in the dependent variable, which is where non-physical oscillations occur. Naturally, as the mesh is refined, less artificial viscosity is needed to produce a smooth solution. Von Neumann and Richtmyer (VonNeumann and Richtmyer, 1950) introduced this method by includ- ing a nonlinear viscosity term with a tuning parameter that was identical to a second-order truncation error.

The original work of Von Neumann and Richtmyer (VonNeumann and Richtmyer, 1950) has seen many improvements and extensions. Tadmor (Tadmor, 1990) examined the use of artificial viscosity in the context of spectral methods and introduced a method where the artificial viscosity is only included for higher frequencies. Inspired by the work of Tamdor, Calhoun-Lopez and Gunzburger (Calhoun- Lopez and Gunzburger, 2006) developed a method for finite elements based on hierarchical basis functions, which allows artificial viscosity to only be added at smaller scales. With that approach, the introduction of the viscosity does not compromise the accuracy of the solution. Xin and Flaherty (Xin and Flaherty, 2006) introduced artificial viscosity at shocks with discontinuous Galerkin finite ele- ments, where the discontinuous Galerkin residual was used to scale the amount of artificial viscosity. Their approach is dependent on a tuning parameter that can be determineda priorifor some systems.

The entropy viscosity (EV) method proposed by Guermond et al (Guermond and Nazarov, 2014; Guermond et al., 2018, 2010, 2017, 2014, 2011; Guermond and Popov, 2014, 2017) is an approach to parameterize the amount of artificial viscosity needed to produce a non-oscillatory solution based on the entropy production of the system and has seen significant developments in recent years. As origi- nally formulated in (Guermond et al., 2011), the entropy viscosity approach is a higher order approach for continuous finite elements that does not depend on flux or slope limiters. The original formulation used the weak form of the Laplacian, which (Guermond and Nazarov, 2014) does not guarantee the maximum principle is obeyed. The entropy viscosity was dependent on two tuning parameters, the lo- cal mesh size, and the entropy, which was normalized by the global entropy average (Guermond et al., 2011). Guermond and Nazarov (Guermond and Nazarov, 2014) improved the EV approach by formu- lating an explicit second-order, maximum-preserving scheme for arbitrary meshes, any Lipschitz flux, and any spatial dimension. This was accomplished by introducing a graph Laplacian for the viscosity, introducing a correction term to the lumped mass matrix to approximate the inverse of a consistent mass matrix, and using the flux-corrected transport paradigm. The entropy viscosity in this updated approach is no longer dependent on the mesh size and includes an edge-based stabilization term but still includes two tuning parameters (Guermond and Nazarov, 2014).

A recent formulation by Guermond et al. (Guermond et al., 2018) is a second-order, parameter- free, edge-based viscosity approach. This approach first determines a maximum-preserving artificial viscosity by either a local-extremum-diminishing approach or a guaranteed maximum-speed approach. The guaranteed maximum-speed approach is preferred because the local-extremum-diminishing ap- proach can violate the entropy condition for transonic rarefactions (Guermond et al., 2018). An ex- tension to the low-order viscosity was developed where the viscosities are smoothed to mimic the flux-limiting approach, while still maintaining the maximum principle (Guermond and Popov, 2017). The higher order viscosity is constructed by taking the minimum of the low-order viscosity and the nondimensional entropy residual, which is a function of the difference between the current solution and the Galerkin solution (Guermond et al., 2018). The entropy residual, as posited in (Guermond and Popov, 2017), includes the time discretization between the Galerkin and current solution, so a “com- mutator” approach was proposed in (Guermond et al., 2018) so the choice of time discretization does not impact the entropy residual.

TCAT is an approach that can be used to formulate continuum mechanical models based upon mi- croscale principles at some combination of larger length scales in each spatial dimension, which may be macroscale or megascale (Gray and Miller, 2014). The microscale is the smallest scale at which continuum mechanical approaches are applicable and applied to a given entity (phase, interface, com- mon curve, common point), and the boundaries of juxtaposed entities are fully resolved in space and time. The macroscale is a length scale at which a point represents the centroid of an averaging region that may contain all entities in some measure. At the macroscale, the boundaries of entities are not explicitly resolved, but measures of the average extent of the entities are evolved (volume fractions, specific interfacial areas, etc). The megascale is the system scale and all phenomena are resolved only through transport through the boundaries of the domain; the entire domain is treated in an averaged sense with no spatial resolution in any megascale dimension. Larger scale three-dimensional domains may be modeled with any combination of macroscale and megascale approaches. Larger scale ap- proaches are necessary for essentially all natural porous medium systems, and many engineered and organismic system as well, due to the computationally intractable nature of microscale approaches for such applications.

The TCAT approach has several attractive features (Battiato et al., 2019; Gray and Miller, 2005, 2014; Miller and Gray, 2005; Miller et al., 2017). These features include a uniform microscale basis for all conservation and thermodynamics, a precise upscaling of all quantities from the microscale to larger scales, the inclusion of interfaces, common curves, and common points, the formulation and use of entropy inequalities to constrain closure relations to be consistent with the second law of thermodynamics, the development and use of evolution equations based on averaging theorems to reduce the set of closure relations needed, geometrically based state equations that are hysteretic free (Miller et al., 2019a), and, because of the rigorous connection between scales, the ability to evaluate and validate larger scale models unambiguously with microscale observations or simulations (Gray et al., 2015). While other upscaling approaches are available (Battiato et al., 2019; Gray et al., 2013), no other upscaling method provides the entire desirable set of attributes annotated above for TCAT.

The TCAT approach involves the following steps. Conservation of mass, momentum, and energy equations, balance of entropy equations, thermodynamic equations, and equilibrium conditions are developed at the microscale for each entity in the system. Averaging operators are applied to each equation and multiscale averaging theorems (Gray et al., 1993; Gray and Miller, 2013, 2014; Miller

and Gray, 2008) are applied to transform the larger scale equations into forms containing the fewest number of variables possible; this transformation results in differentials of averaged quantities rather than averages of differential quantities. A larger scale entropy inequality is developed for the system and connected to the dissipative processes that produce entropy, which are specified in the conserva- tion equations. A resultant flux-force form is developed for entropy production, which is in turn used to constrain the permissible form of closure relations that can be used to formulate a closed, solvable model at the larger scale. Evolution equations, derived purely from the averaging theorems (Gray et al., 2015; Gray and Miller, 2010, 2014), and equations of state (Gray et al., 2019; McClure et al., 2018; Miller et al., 2019a) also are used to resolve the closure problem and produce closed, solvable models for which all quantities are described in terms of microscale precursors. The TCAT approach has been used to derive macroscale models for single-fluid flow (Gray and Miller, 2006), single-fluid flow and species transport (Gray and Miller, 2009; Miller and Gray, 2008; Weigand et al., 2018b), two-fluid flow (Jackson et al., 2009), and two-fluid flow and species transport (Rybak et al., 2015), all for porous medium systems. TCAT model hierarchies have also been developed for single fluid flow in a porous medium at the megascale (Gray and Miller, 2009), the transition between a two-fluid-phase porous medium system and a single-fluid system (Jackson et al., 2012), and for sediment transport in turbulent surface waters (Miller et al., 2018a, 2019b).

The TCAT entropy inequality yields an expression for the rate of entropy production resulting from dissipative processes (Gray and Miller, 2014; Miller and Gray, 2005). This physically-based entropy production rate could be of use in formulating artificial dissipation in numerical methods, such as the EV method. Such an approach has not been applied or evaluated.