The moving discontinuous Galerkin finite element method with interface condition enforcement for compressible viscous flows

Abstract

The moving discontinuous Galerkin finite element method with interface condition enforcement is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or underresolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear r‐adaptivity. This approach avoids introducing low‐order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier–Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of 105 in two‐dimensional space. Time accurate solutions of unsteady problems are obtained via a space‐time formulation, in which the unsteady problem is formulated as a higher dimensional steady space‐time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization.

A robust, high-order implicit shock tracking method for simulation of complex, high-speed flows

“…As a result, the method produces highly accurate solutions on coarse meshes and recovers optimal convergence rates of the DG discretization even for flows with non-smooth features. While these advantages are shared by explicit shock tracking approaches [29,35], HOIST [43,46,45] and other implicit shock tracking methods (e.g., MDG-ICE [9,23,24]) provide several additional benefits: formulated for a general, nonlinear conservation law (not tailored to a specific set of equations) and the formulation is well-suited for problems with intricate discontinuity surfaces (curved and reflecting shocks, shock formation, shock-shock interaction). The latter is due to the geometrically complex problem of generating a featurealigned mesh being re-cast to solving an optimization problem over the discrete DG solution and nodal coordinates of the mesh.…”

“…. , du into unconstrained and constrained degrees of freedom is not unique; the only condition is B I Y I be non-singular because the RanpY I q ' RanpZ I q " R d condition is guaranteed by construction in (24). We choose the unconstrained degrees of freedom xu I to correspond to the coordinate directions closest to the null space of B I , denoted NullpB I q, because this will require the least action on the constrained degrees of freedom.…”

“…A new class of high-order numerical methods has recently emerged, implicit shock tracking, that includes the High-Order Implicit Shock Tracking (HOIST) method [43,46,45] and the Moving Discontinous Galerkin Method with Interface Condition Enforcement (MDG-ICE) [9,23,24]. Like traditional shock tracking, these methods align element faces with discontinuity surfaces to represent them perfectly and the high-order basis functions approximate the smooth solution away from shocks.…”

On the Origins of Lagrangian Hydrodynamic Methods

“…al. 35,36 The schemes by Peierls and Skyrme were never published in a journal so they remained somewhat lost to time and unknown to the broader scientific community. The details on Skyrme's shock-fitting scheme were documented in multiple laboratory reports and letters during and after the Manhattan Project.…”

UnDiFi-2D: an Unstructured Discontinuity Fitting code for 2D grids

“…Shock-fitting/front-tracking ideas made their way also through the Finite Element community. We refer to the SUPG technique of [41] and the Discontinuous Galerkin (DG) Finite Element methods (FEM) independently developed by two different research teams: [42,43] and [44,45]. All three aforementioned techniques simultaneously solve for the location of the grid-points, in addition to the flow-variables, so as to constrain certain edges of the tessellation to be aligned with the discontinuities.…”