350 citations · 908 across the 39 of their papers we have counts for
45 papers
Summation-by-parts operators for general function spaces: Flux differencing, entropy stability, and accuracy
Henry M. Haase, Andrew R. Winters, Jan Glaubitz
Summation-by-parts (SBP) operators are a cornerstone of provably stable high-order methods for hyperbolic conservation laws. For nonlinear problems, the flux-differencing framework…
Well-Balanced Subcell Limiting for Discontinuous Galerkin Discretizations of the Shallow-Water Equations
Andrés M. Rueda-Ramírez, Patrick Ersing, Andrew R. Winters +1
High-order discontinuous Galerkin (DG) methods equipped with subcell finite-volume (FV) limiters provide an efficient framework for the simulation of nonlinear hyperbolic balance l…
A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extension
Philipp Öffner, Per Pettersson, Andrew R. Winters
Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. Bender and Öffner developed entropy-conservat…
Entropy analysis and entropy stable DG methods for the 1D shallow water moment equations
Julio Careaga, Patrick Ersing, Julian Koellermeier +1
We demonstrate that the one-dimensional shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Ad…
Boundary-optimized closures for diagonal-norm upwind SBP operators
Ken Mattsson, David Niemelä, Andrew R. Winters
By employing non-equispaced grid points near boundaries, boundary-optimized upwind finite-difference operators of orders up to nine are developed. The boundary closures are constru…
A new class of entropy stable fluctuations for the discontinuous Galerkin method with application to the Saint-Venant-Exner model
Patrick Ersing, Andrew R. Winters
In this work we consider entropy stable discontinuous Galerkin methods applied to nonconservative hyperbolic systems. We introduce a new class of entropy conservative fluctuations…
Numerical boundary flux functions that give provable bounds for nonlinear initial boundary value problems with open boundaries
Andrew R. Winters, David A. Kopriva, Jan Nordström
We present a strategy for interpreting nonlinear, characteristic-type penalty terms as numerical boundary flux functions that provide provable bounds for solutions to nonlinear hyp…
Towards provable energy-stable overset grid methods using sub-cell summation-by-parts operators
Jan Glaubitz, Joshua Lampert, Andrew R. Winters +1
Overset grid methods handle complex geometries by overlapping simpler, geometry-fitted grids to cover the original, more complex domain. However, ensuring their stability---particu…
Global Bounds for the Error in Solutions of Linear Hyperbolic Systems due to Inaccurate Boundary Geometry
David A. Kopriva, Andrew R. Winters, Jan Nordström
Meshes approximate the boundaries of a geometry when the boundaries are curved. The accuracy of the mesh then affects the error of computations of initial boundary value problems f…
Entropy stable hydrostatic reconstruction schemes for shallow water systems
Patrick Ersing, Sven Goldberg, Andrew R. Winters
In this work, we develop a new hydrostatic reconstruction procedure to construct well-balanced schemes for one and multilayer shallow water flows, including wetting and drying. Ini…
Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods
Jan Glaubitz, Hendrik Ranocha, Andrew R. Winters +3
High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly fo…
Energy Bounds for Discontinuous Galerkin Spectral Element Approximations of Well-Posed Overset Grid Problems for Hyperbolic Systems
David A. Kopriva, Andrew R. Winters, Jan Nordström
We show that even though the Discontinuous Galerkin Spectral Element Method is stable for hyperbolic boundary-value problems, and the overset domain problem is well-posed in an app…
On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
Hendrik Ranocha, Andrew R. Winters, Michael Schlottke-Lakemper +3
We use the framework of upwind summation-by-parts (SBP) operators developed by Mattsson (2017, doi:10.1016/j.jcp.2017.01.042) and study different flux vector splittings in this con…
An entropy stable discontinuous Galerkin method for the two-layer shallow water equations on curvilinear meshes
Patrick Ersing, Andrew R. Winters
We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic…
Encapsulated generalized summation-by-parts formulations for curvilinear and non-conforming meshes
Tomas Lundquist, Andrew Winters, Jan Nordström
We extend the construction of so-called encapsulated global summation-by-parts operators to the general case of a mesh which is not boundary conforming. Owing to this development,…
On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics
Hendrik Ranocha, Andrew R. Winters, Hugo Guillermo Castro +4
We study temporal step size control of explicit Runge-Kutta methods for compressible computational fluid dynamics (CFD), including the Navier-Stokes equations and hyperbolic system…
Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws
Hendrik Ranocha, Michael Schlottke-Lakemper, Jesse Chan +4
Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entro…
Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing
Hendrik Ranocha, Michael Schlottke-Lakemper, Andrew R. Winters +3
We present Trixi.jl, a Julia package for adaptive high-order numerical simulations of hyperbolic partial differential equations. Utilizing Julia's strengths, Trixi.jl is extensible…
An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
Andrés M Rueda-Ramírez, Sebastian Hennemann, Florian J Hindenlang +2
The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible m…
A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics
Michael Schlottke-Lakemper, Andrew R. Winters, Hendrik Ranocha +1
One of the challenges when simulating astrophysical flows with self-gravity is to compute the gravitational forces. In contrast to the hyperbolic hydrodynamic equations, the gravit…
Stable filtering procedures for nodal discontinuous Galerkin methods
Jan Nordström, Andrew R. Winters
We prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomi…
Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier-Stokes Equations
Andrew R. Winters, David A. Kopriva, Gregor J. Gassner +1
Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order…
Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Navier-Stokes Equations
David C. Del Rey Fernandez, Mark H. Carpenter, Lisandro Dalcin +4
The entropy conservative, curvilinear, nonconforming, p-refinement algorithm for hyperbolic conservation laws of Del Rey Fernandez et al. (2019), is extended from the compressible…
Entropy Stable p-Nonconforming Discretizations with the Summation-by-Parts Property for the Compressible Euler equations
D. C. Del Rey Fernandez, M. H. Carpenter, L. Dalcin +6
The entropy conservative/stable algorithm of Friedrich~\etal (2018) for hyperbolic conservation laws on nonconforming p-refined/coarsened Cartesian grids, is extended to curvilinea…