Mark H. Carpenter

K. Mattsson, Mark H. Carpenter

Block-to-block interface interpolation operators are constructed for several common high-order finite difference discretizations. In contrast to conventional interpolation operators, these new interpolation operators maintain the strict stability, accuracy and conservation of the base scheme even when nonconforming grids or dissimilar operators are used in adjoining blocks. The stability properties of the new operators are verified using eigenvalue analysis, and the accuracy properties are verified using numerical simulations of the Euler equations in two spatial dimensions.

Lucas Friedrich, Gero Schnücke, Andrew R. Winters et al.

This work examines the development of an entropy conservative (for smooth solutions) or entropy stable (for discontinuous solutions) space-time discontinuous Galerkin (DG) method for systems of non-linear hyperbolic conservation laws. The resulting numerical scheme is fully discrete and provides a bound on the mathematical entropy at any time according to its initial condition and boundary conditions. The crux of the method is that discrete derivative approximations in space and time are summation-by-parts (SBP) operators. This allows the discrete method to mimic results from the continuous entropy analysis and ensures that the complete numerical scheme obeys the second law of thermodynamics. Importantly, the novel method described herein does not assume any exactness of quadrature in the variational forms that naturally arise in the context of DG methods. Typically, the development of entropy stable schemes is done on the semi-discrete level ignoring the temporal dependence. In this work we demonstrate that creating an entropy stable DG method in time is similar to the spatial discrete entropy analysis, but there are important (and subtle) differences. Therefore, we highlight the temporal entropy analysis throughout this work. For the compressible Euler equations, the preservation of kinetic energy is of interest besides entropy stability. The construction of kinetic energy preserving (KEP) schemes is, again, typically done on the semi-discrete level similar to the construction of entropy stable schemes. We present a generalization of the KEP condition from Jameson to the space-time framework and provide the temporal components for both entropy stability and kinetic energy preservation. The properties of the space-time DG method derived herein is validated through numerical tests for the compressible Euler equations.

Lucas Friedrich, Andrew R. Winters, David C. Del Rey Fernández et al.

This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of non-linear conservation laws with general geometric (h) and polynomial order (p) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between nonconforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for non-linear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general non-linear hyperbolic systems on h/p non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

Nail K. Yamaleev, David C. Del Rey Fernandez, Jialin Lou et al.

New entropy stable spectral collocations schemes of arbitrary order of accuracy are developed for the unsteady 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. To take into account the grid motion and deformation, we use an arbitrary Lagrangian-Eulerian (ALE) formulation. As a result, moving and deforming hexahedral grid elements are individually mapped onto a cube in the fixed reference system of coordinates. The proposed scheme is constructed by using the skew-symmetric form of the Navier-Stokes equations, which are discretized by using summation-by-parts spectral collocation operators that preserve the conservation properties of the original governing equations. Furthermore, the metric coefficients are approximated such that the geometric conservation laws (GCL) are satisfied exactly on both static and dynamic grids. To make the scheme entropy stable, a new entropy conservative flux is derived for the 3-D Euler and Navier-Stokes equations on dynamic unstructured grids. The new flux preserves the design order of accuracy of the original spectral collocation scheme and guarantees the entropy conservation on moving and deforming grids. We present numerical results demonstrating design order of accuracy and freestream preservation properties of the new schemes for both the Euler and Navier-Stokes equations on moving and deforming unstructured grids.