Andrew R. Winters

Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations

Fisher and Carpenter (2013) [12] found a remarkable equivalence of general diagonal norm high-order summation-by-parts operators to a subcell based high-order finite volume formulation. This equivalence enables the construction of provably entropy stable schemes by a specific choice of the subcell finite volume flux. We show that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms. Thus, by changing the subcell finite volume flux to a specific choice, we are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms, such as the Ducros splitting and the Kennedy and Gruber splitting. Although these split forms are not entropy stable, we present a systematic way to prove which of those split forms are at least kinetic energy preserving. With this, we construct a unified high-order split form DG framework. We investigate with three dimensional numerical simulations of the inviscid TaylorGreen vortex and show that the new split forms enhance the robustness of high-order simulations in comparison to the standard scheme when solving turbulent vortex dominated flows. In fact, we show that for certain test cases, the novel split form discontinuous Galerkin schemes are more robust than the discontinuous Galerkin scheme with over-integration.

A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations

In this work, we design an arbitrary high order accurate nodal discontinuous Galerkin spectral element type method for the one dimensional shallow water equations. The novel method uses a skew-symmetric formulation of the continuous problem. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, we show that combined with a special numerical interface flux function, the method exactly preserves the entropy, which is also the total energy for the shallow water equations. Finally, we prove that the surface fluxes, the skew-symmetric volume integrals, and the source term are well balanced. Numerical tests are performed to demonstrate the theoretical findings.

Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations

In this work, we design an entropy stable, finite volume approximation for the ideal magnetohydrodynamics (MHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that discretely preserves the entropy of the system. To guarantee the discrete conservation of entropy requires the addition of a particular source term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate energy at shocks, thus to compute accurate solutions to problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.

An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the nonlinear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretisation exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretisation of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

The BR1 Scheme is Stable for the Compressible Navier–Stokes Equations

In this work we prove that the original (Bassi and Rebay in J Comput Phys 131:267–279, 1997) scheme (BR1) for the discretization of second order viscous terms within the discontinuous Galerkin collocation spectral element method (DGSEM) with Gauss Lobatto nodes is stable. More precisely, we prove in the first part that the BR1 scheme preserves energy stability of the skew-symmetric advection term DGSEM discretization for the linearized compressible Navier–Stokes equations (NSE). In the second part, we prove that the BR1 scheme preserves the entropy stability of the recently developed entropy stable compressible Euler DGSEM discretization of Carpenter et al. (SIAM J Sci Comput 36:B835–B867, 2014) for the non-linear compressible NSE, provided that the auxiliary gradient equations use the entropy variables. Both parts are presented for fully three-dimensional, unstructured curvilinear hexahedral grids. Although the focus of this work is on the BR1 scheme, we show that the proof naturally includes the Local DG scheme of Cockburn and Shu.

A novel high-order, entropy stable, 3D AMR MHD solver with guaranteed positive pressure

We describe a high-order numerical magnetohydrodynamics (MHD) solver built upon a novel non-linear entropy stable numerical flux function that supports eight travelling wave solutions. By construction the solver conserves mass, momentum, and energy and is entropy stable. The method is designed to treat the divergence-free constraint on the magnetic field in a similar fashion to a hyperbolic divergence cleaning technique. The solver described herein is especially well-suited for flows involving strong discontinuities. Furthermore, we present a new formulation to guarantee positivity of the pressure. We present the underlying theory and implementation of the new solver into the multi-physics, multi-scale adaptive mesh refinement (AMR) simulation code FLASH (http://flash.uchicago.edu). The accuracy, robustness and computational efficiency is demonstrated with a number of tests, including comparisons to available MHD implementations in FLASH.

High-Order Local Time Stepping on Moving DG Spectral Element Meshes

We derive and evaluate an explicit local time stepping (LTS) integration for the discontinuous Galerkin spectral element method on moving meshes. The LTS procedure is derived from Adams–Bashforth multirate time integration methods. We also present speedup and memory estimates, which show that the explicit LTS integration scales well with problem size. Time-step refinement studies with static and moving meshes show that the approximations are spectrally accurate in space and have design temporal accuracy. The numerical tests validate theoretical estimates that the LTS procedure can reduce computational cost by as much as an order of magnitude for time accurate problems.

A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations

We describe a unique averaging procedure to design an entropy stable dissipation operator for the ideal magnetohydrodynamic (MHD) and compressible Euler equations. Often in the derivation of an ent ...

A novel averaging technique for discrete entropy-stable dissipation operators for ideal MHD

Entropy stable schemes can be constructed with a specific choice of the numerical flux function. First, an entropy conserving flux is constructed. Secondly, an entropy stable dissipation term is added to this flux to guarantee dissipation of the discrete entropy. Present works in the field of entropy stable numerical schemes are concerned with thorough derivations of entropy conservative fluxes for ideal MHD. However, as we show in this work, if the dissipation operator is not constructed in a very specific way, it cannot lead to a generally stable numerical scheme.The two main findings presented in this paper are that the entropy conserving flux of Ismail & Roe can easily break down for certain initial conditions commonly found in astrophysical simulations, and that special care must be taken in the derivation of a discrete dissipation matrix for an entropy stable numerical scheme to be robust.We present a convenient novel averaging procedure to evaluate the entropy Jacobians of the ideal MHD and the compressible Euler equations that yields a discretization with favorable robustness properties.

A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography

In this work, we compare and contrast two provably entropy stable and high-order accurate nodal discontinuous Galerkin spectral element methods applied to the one dimensional shallow water equations for problems with non-constant bottom topography. Of particular importance for numerical approximations of the shallow water equations is the well-balanced property. The well-balanced property is an attribute that a numerical approximation can preserve a steady-state solution of constant water height in the presence of a bottom topography. Numerical tests are performed to explore similarities and differences in the two high-order schemes.