Adaptive spacetime discontinuous Galerkin method for hyperbolic advection-diffusion with a non-negativity constraint

Abstract

Applications where the diffusive and advective time scales are of similar order give rise to advectiondiffusion phenomena that are inconsistent with the predictions of parabolic Fickian diffusion models. Non-Fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in Fickian models. In this work, we implement a modified, frame-invariant form of Cattaneo's hyperbolic diffusion relation within a spacetime discontinuous Galerkin advection-diffusion model. An hadaptive spacetime meshing procedure supports an asynchronous, patch-by-patch solution procedure with linear computational complexity in the number of spacetime elements. This localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non-negative concentration solution. In contrast to some previous methods, we do not modify the numerical fluxes to enforce non-negative concentrations. Thus, the element-wise conservation properties that are intrinsic to discontinuous Galerkin models are defined with respect to physically meaningful Riemann fluxes on the element boundaries. We present numerical examples that demonstrate the effectiveness of the proposed model, and we explore the distinct features of hyperbolic advection-diffusion response in subcritical and supercritical flows. Ã q D Kr ;and show that the resulting hyperbolic system exhibits finite propagation velocities and is Galileaninvariant. We refer to (5) as the modified Maxwell-Cattaneo (mMC) model and use it as our reference hyperbolic model for advection-diffusion from here on. This paper is concerned with a finite element method for solving hyperbolic advection-diffusion problems, such as (5). A survey of the literature on numerical methods that address hyperbolic diffusion, for both pure diffusion [14,[19][20][21], and combined advection-diffusion [1,22,23], can be found in [24]. For the pure diffusion case, numerical models based on discontinuous Galerkin (DG) finite element methods, [22,25], are most relevant to the method proposed here. In particular, the adaptive spacetime DG method described in [26] is a direct antecedent to the method advanced in this work. For the mMC advection-diffusion problem, the DG method in [22] is the closest HYPERBOLIC ADVECTION-DIFFUSION WITH NON-NEGATIVITY CONSTRAINT Unconditional stability for linear systems, such as (5). Falk and Richter prove stability of SDG solutions for linear, symmetric hyperbolic systems in [44], and Lowrie et al. report similar findings in [43] for linear systems of conservation laws, such as the one in this work. Compact computational stencils that do not expand with higher-order approximations; cf. discussion in first paragraph in Section 4. Element-wise conservation with respect to computed Riemann fluxes.

Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations

“…The GAL and SUPG formulations do not possess this property without any further modification or enrichment to their formulations. One way to ensure this property under a single-field finite element framework is through the use of the DG formulations (see [Arnold et al, 2002;Riviére and Wheeler, 2002;Cockburn, 2003;Riviére, 2015b,a, 2016;Pal et al, 2016] and the references within for further details). To present the DG formulation employed in the paper, we now introduce relevant notation.…”

“…A comprehensive list and discussion of other prior works related to enforcing mesh restrictions to meet the maximum principle and the non-negative constraint can also be found in [Mudunuru and Nakshatrala, 2016b]. (c) Developing or altering formulations in the continuum setting: Two works that fall under this category are [Harari, 2004;Pal et al, 2016], both of which addressed transient transport problems. [Harari, 2004] utilized a stabilized method that is available for Helmholtz-type equations to construct a stabilized formulation for transient isotropic diffusion equations to meet the maximum principle.…”

“…This approach, as presented in [Harari, 2004], is applicable to one-dimensional setting. [Pal et al, 2016] meets maximum principles for transient transport equations by employing two techniques. They rewrote transient transport equations, which are parabolic in nature, into modified Maxwell-Cattaneo equations, which are hyperbolic in nature, and employed the space-time Discontinuous Galerkin approach.…”

“…The discrete formulation is also symmetric and positive-definite, so one can easily apply both bounded constraints and equality constraints to ensure non-negative solutions and local mass conservation respectively. It should be noted that one may also employ normal equations or the least-squares approach to ensure that the minimization problem for non-symmetric problems is convex [Demmel, 1997;Burdakov et al, 2012;Pal et al, 2016;Chang et al, 2016b]. All of these studies have employed quadratic programming (QP) techniques to enforce the maximum principle on 2D academic problems, but the problems studied are small-scale and do not require state-of-the-art Krylov subspace (KSP) iterative solvers and preconditioners.…”

A stabilized mixed discontinuous Galerkin formulation for double porosity/permeability model

“…One of the first successful applications of DG formulation to solve a practical problem was by [Reed and Hill, 1973], which addressed neutron transport. Over the years, DG methods have been successfully employed to solve hyperbolic PDEs [Brezzi et al, 2004;Pal et al, 2016], elliptic PDEs [Arnold et al, 2002;Barrios and Bustinzal, 2007;Cockburn et al, 2009b;Douglas and Dupont, 1976;Rivière et al, 1999;Rusten et al, 1996], parabolic PDEs [Douglas and Dupont, 1976;Kulkarni et al, 2007], coupling algorithms [Nakshatrala et al, 2009] and space-time finite elements [Abedi et al, 2006;Palaniappan et al, 2004]. Several variants of DG formulations have been developed over the years with varying merits for each variant.…”