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High-order Discontinuous Galerkin Spectral Element Methods (DGSEM) provide excellent accuracy for complex flow simulations, but their computational cost increases sharply with higher polynomial orders. %that provide very accurate solutions.…

Fluid Dynamics · Physics 2025-12-11 Xukun Wang , Oscar A. Marino , Esteban Ferrer

In the present work, we extend the Discontinuous Galerkin Spectral Element Method (DGSEM) to high-enthalpy reacting gas flows with internal degrees of freedom. An entropy- and kinetic energy-preserving flux function is proposed which allows…

Fluid Dynamics · Physics 2024-11-21 Georgii Oblapenko , Arseniy Tarnovskiy , Moritz Ertl , Manuel Torrilhon

Hydrodynamical numerical methods that converge with high-order hold particular promise for astrophysical studies, as they can in principle reach prescribed accuracy goals with higher computational efficiency than standard second- or…

Instrumentation and Methods for Astrophysics · Physics 2023-05-05 Miha Cernetic , Volker Springel , Thomas Guillet , Rüdiger Pakmor

Unstructured-mesh ocean models are increasingly used for coastal applications due to their ability to represent complex geometries and apply local grid refinement where needed. However, their broader use has been hindered by their high…

Distributed, Parallel, and Cluster Computing · Computer Science 2026-05-18 Miguel De Le Court , Vincent Legat , Ange P. Ishimwe , Colin Scherpereel , Emmanuel Hanert , Jonathan Lambrechts

Hyperbolic-parabolic partial differential equations are widely used for the modeling of complex, multiscale problems. High-order methods such as the discontinuous Galerkin (DG) scheme are attractive candidates for their numerical…

Numerical Analysis · Mathematics 2025-07-08 Jens Keim , Anna Schwarz , Patrick Kopper , Marcel Blind , Christian Rohde , Andrea Beck

We discuss the development, verification, and performance of a GPU accelerated discontinuous Galerkin method for the solutions of two dimensional nonlinear shallow water equations. The shallow water equations are hyperbolic partial…

Numerical Analysis · Mathematics 2023-07-19 R Gandham , D S Medina , T Warburton

High order (HO) schemes are attractive candidates for the numerical solution of multiscale problems occurring in fluid dynamics and related disciplines. Among the HO discretization variants, discontinuous Galerkin schemes offer a collection…

Computational Engineering, Finance, and Science · Computer Science 2019-10-08 Nico Krais , Andrea Beck , Thomas Bolemann , Hannes Frank , David Flad , Gregor Gassner , Florian Hindenlang , Malte Hoffmann , Thomas Kuhn , Matthias Sonntag , Claus-Dieter Munz

Accurate and efficient simulation of fluid-structure interaction (FSI) problems remains a central challenge in computational physics. High-order discontinuous Galerkin (DG) methods offer low numerical errors and excellent scalability on…

Fluid Dynamics · Physics 2025-12-08 Yingjie Xia , Stefano Colombo , David Huergo , Jiaqing Kou , Yuting Dai , Esteban Ferrer

We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are…

Numerical Analysis · Mathematics 2020-01-08 Hong Xiao , Eky Febrianto , Qiaoling Zhang , Fehmi Cirak

High-order discontinuous Galerkin spectral element methods (DGSEM) have received growing attention and development, especially in the regime of computational fluid dynamics in recent years. The inherent flexibility of the discontinuous…

Numerical Analysis · Mathematics 2025-12-22 Boyang Xia , David Moxey

We present a high-order implicit large eddy simulation (ILES) approach for simulating flows at the nearly incompressible regime. Our methodology based on utilization of a nodal discontinuous Galerkin (DG) discretization of the Boltzmann…

Fluid Dynamics · Physics 2026-01-05 Onur Ata , Atakan Aygun , Tim Warburton , Ali Karakus

We present a high-order implicit large-eddy simulation (ILES) approach for simulating transitional turbulent flows. The approach consists of an Interior Embedded Discontinuous Galerkin (IEDG) method for the discretization of the…

Fluid Dynamics · Physics 2016-05-04 Pablo Fernandez , Ngoc-Cuong Nguyen , Xevi Roca , Jaime Peraire

High-order accurate discontinuous Galerkin (DG) methods have emerged as powerful tools for solving partial differential equations such as the compressible Navier-Stokes equations due to their excellent dispersion-dissipation properties and…

Numerical Analysis · Mathematics 2025-11-12 Patrick Kopper , Anna Schwarz , Jens Keim , Andrea Beck

Implicit methods and GPU parallelization are two distinct yet powerful strategies for accelerating high-order CFD algorithms. However, few studies have successfully integrated both approaches within high-speed flow solvers. The core…

Numerical Analysis · Mathematics 2025-09-09 Hongyu Liu , Xing Ji , Yuan Fu , Kun Xu

In this study, three-dimensional simulations of a reacting hydrogen jet in supersonic crossflow using a structure-preserving discontinuous Galerkin (DG) formulation are examined. The hydrogen jet, with a momentum flux ratio of five, is…

Fluid Dynamics · Physics 2025-08-13 Cal J. Rising , Eric J. Ching , Ryan F. Johnson

We present a GPU-accelerated version of a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier-Stokes equations. The equations are discretized in time using a semi-implicit scheme with explicit treatment of…

Numerical Analysis · Mathematics 2018-05-08 Ali Karakus , Noel Chalmers , Kasia Swirydowicz , Timothy Warburton

This paper presents heavily grad-div and pressure jump stabilised, equal- and mixed-order discontinuous Galerkin finite element methods for non-isothermal incompressible flows based on the Oberbeck-Boussinesq approximation. In this…

Numerical Analysis · Mathematics 2017-08-16 Philipp W. Schroeder , Gert Lube

The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to…

Numerical Analysis · Mathematics 2024-10-15 Eike Hermann Müller

A discontinuous Galerkin method for the discretization of the compressible Euler equations, the governing equations of inviscid fluid dynamics, on Cartesian meshes is developed for use of Graphical Processing Units via OCCA, a unified…

Computational Physics · Physics 2020-09-01 Andrew C. Kirby , Dimitri J. Mavriplis

Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…

Numerical Analysis · Mathematics 2009-11-18 Andreas Klöckner , Tim Warburton , Jeffrey Bridge , Jan S. Hesthaven