We develop an entropy-stable high-order numerical method for the two-dimensional compressible Euler equations on general curvilinear meshes. The proposed approach is based on a nodal discontinuous Galerkin spectral element method (DGSEM) that satisfies the summation-by-parts (SBP) property. At the semidiscrete level, entropy stability is established through the SBP structure and the discrete metric identities associated with curvilinear coordinate mappings. By incorporating entropy-stable numerical fluxes at element interfaces, a global discrete entropy inequality is obtained. To further control nonphysical oscillations near strong discontinuities, the entropy-stable DG formulation is combined with a modified oscillation-eliminating discontinuous Galerkin (OEDG) method, which was originally proposed in [59]. We observe that the zero-order damping coefficient in the original OEDG method naturally serves as an effective shock indicator, which enables localization of the oscillation control mechanism and significantly reduces computational cost. Moreover, while the original OEDG formulation relies on local orthogonal modal bases and is primarily restricted to simplicial meshes, we reformulate the OE procedure using projection operators, allowing for a systematic extension to general curvilinear meshes. The resulting method preserves conservation and entropy stability while effectively suppressing spurious oscillations. A series of challenging numerical experiments is presented to demonstrate the accuracy, robustness, and effectiveness of the proposed entropy-stable OEDG method on both Cartesian and curvilinear meshes.
h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes
Institute of Aerodynamics and Gas Dynamics, University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）
Positivity-Preserving and Entropy-Stable Oscillation-Eliminating DGSEM for the Compressible Euler Equations on Curvilinear Meshes with Adaptive Mesh Refinement
In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms. Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galer…
Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes with Summation-by-Parts Property for Hyperbolic Conservation Laws
University of Cologne（科隆大学）；Institute of Aerodynamics and Gas Dynamics (IAG), University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Center for Data and Simulation Science (CDS), University of Cologne（科隆大学数据与模拟科学中心）
The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment. Hennemann et al. [DOI…
In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton-Jacobi (H-J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme [10] for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme …
This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut [Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhauser] a…
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
Universität zu Köln（科隆大学）；The Florida State University（佛罗里达州立大学）
A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields: Curvilinear coordinates and multi-block domains
Max Plank Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；The University of Texas at El Paso（得克萨斯大学埃尔帕索分校）；Proxima Fusion GmbH（Proxima Fusion有限公司）；Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学研究所）
Symmetry-preserving (mimetic) discretization aims to preserve certain properties of a continuous differential operator in its discrete counterpart. For these discretizations, stability and (discrete) conservation of mass, momentum and energy are proven in the same way as for the original continuous model. This paper presents a new finite-difference symmetry-preserving space discretization. Boundary conditions and time integration are not addressed. The novelty is that it combines arbitrary order of convergence, orthogonal and non-orthogonal structured curvilinear staggered meshes, and the appl…
We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomia…
In this work a non-conservative balance law formulation is considered that encompasses the rotating, compressible Euler equations for dry atmospheric flows. We develop a semi-discretely entropy stable discontinuous Galerkin method on curvilinear meshes using a generalization of flux differencing for numerical fluxes in fluctuation form. The method uses the skew-hybridized formulation of the element operators to ensure that, even in the presence of under-integration on curvilinear meshes, the resulting discretization is entropy stable. Several atmospheric flow test cases in one, two, and three …
When time-dependent partial differential equations (PDEs) are solved numerically in a domain with curved boundary or on a curved surface, mesh error and geometric approximation error caused by the inaccurate location of vertices and other interior grid points, respectively, could be the main source of the inaccuracy and instability of the numerical solutions of PDEs. The role of these geometric errors in deteriorating the stability and particularly the conservation properties are largely unknown, which seems to necessitate very fine meshes especially to remove geometric approximation error. Th…
We develop a fourth order accurate finite difference method for the three dimensional elastic wave equation in isotropic media with the piecewise smooth material property. In our model, the material property can be discontinuous at curved interfaces. The governing equations are discretized in second order form on curvilinear meshes by using a fourth order finite difference operator satisfying a summation-by-parts property. The method is energy stable and high order accurate. The highlight is that mesh sizes can be chosen according to the velocity structure of the material so that computational…
Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；Australian National University（澳大利亚国立大学）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）
The spectral element method constructed by the $Q^k$ ($k\geq 2$) continuous finite element method with $(k+1)$-point Gauss-Lobatto quadrature on rectangular meshes is a popular high order scheme for solving wave equations in various applications. It can also be regarded as a finite difference scheme on all Gauss-Lobatto points. We prove that this finite difference scheme is $(k+2)$-order accurate in discrete 2-norm for smooth solutions. The same proof can be extended to the spectral element method solving linear parabolic and Schrödinger equations. The main result also applies to the spectral …
Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows
Institute of Mathematics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；Institute for Atmospheric Physics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学大气物理研究所）
High order accurate and explicit time-stable solvers are well suited for hyperbolic wave propagation problems. As a result of the complexities of real geometries, internal interfaces and nonlinear boundary and interface conditions, discontinuities and sharp wave fronts may become fundamental features of the solution. Thus, geometrically flexible and adaptive numerical algorithms are critical for high fidelity and efficient simulations of wave phenomena in many applications. Adaptive curvilinear meshes hold promise to minimise the effort to represent complicated geometries or heterogeneous mate…
The recently proposed high-order TENO scheme [Fu et al., Journal of Computational Physics, 305, pp.333-359] has shown great potential in predicting complex fluids owing to the novel weighting strategy, which ensures the high-order accuracy, the low numerical dissipation, and the sharp shock-capturing capability. However, the applications are still restricted to simple geometries with Cartesian or curvilinear meshes. In this work, a new class of high-order shock-capturing TENO schemes for unstructured meshes are proposed. Similar to the standard TENO schemes and some variants of WENO schemes, t…
A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each vertex is computed directly from fluxes nearby by means of high order least-square. In order to capture discontinuities, numerical flux of high order accuracy is calculated on each edge and serves as supporting data of the least-square computation of the divergence. The high accuracy of the numerical flux depends on the high order WENO interpolation on each ed…
This work presents a high-accuracy, mesh-free, generalized Stokes theorem-based numerical quadrature scheme for integrating functions over trimmed parametric surfaces and volumes. The algorithm relies on two fundamental steps: (1) We iteratively reduce the dimensionality of integration using the generalized Stokes theorem to line integrals over trimming curves, and (2) we employ numerical antidifferentiation in the generalized Stokes theorem using high-order quadrature rules. The scheme achieves exponential convergence up to trimming curve approximation error and has applications to computatio…
A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
ETSIAE-UPM - School of Aeronautics, Universidad Politécnica de Madrid（马德里理工大学航空学院）；Center for Computational Simulation, Universidad Politécnica de Madrid（马德里理工大学计算模拟中心）；Department of Mathematics, Florida State University（佛罗里达州立大学数学系）；Computational Science Research Center, San Diego State University（圣地亚哥州立大学计算科学研究中心）
We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation…
Second order finite volume scheme for Euler equations with gravity which is well-balanced for general equations of state and grid systems
Univ. of Würzburg（维尔茨堡大学）；TIFR Center for Applicable Mathematics（TIFR应用数学中心）；Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik（海德堡大学天文学中心，理论天体物理研究所）；Heidelberger Institut für Theoretische Studien（海德堡理论研究所）
High-order accurate summation-by-parts (SBP) finite difference (FD) methods constitute efficient numerical methods for simulating large-scale hyperbolic wave propagation problems. Traditional SBP FD operators that approximate first-order spatial derivatives with central-difference stencils often have spurious unresolved numerical wave-modes in their computed solutions. Recently derived high order accurate upwind SBP operators based upwind FD stencils have the potential to suppress these poisonous spurious wave-modes on marginally resolved computational grids. In this paper, we demonstrate that…
A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The…
