Central Discontinuous Galerkin Research Articles

GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), meaning they consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by the BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system, which is the relativistic analogue of Godunov's symmetrizable form of the non-relativistic MHD system (Godunov, 1972 [19]). A key novelty in our schemes is the meticulous discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. Notably, we provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework (Wu and Shu, 2023 [49]), which equivalently converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. Finally, we implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios like ultra-relativistic blasts and jets in strongly magnetized environments.

Central discontinuous Galerkin finite element method for the time-fractional convection equation in two space dimensions

In this study, an efficient approach for solving the two-dimensional time-fractional convection problem with a non-smooth solution in the temporal direction is proposed. The solution exhibits weakly regular at the initial time, so the L1 formula on the nonuniform meshes is applied to discretizing the Caputo derivative. In the spatial direction, the central discontinuous Galerkin (CDG) method involving two approximate solutions defined on overlapping elements is used. The fully discrete scheme is proven to be numerically stable, and the optimal convergence result is attained. A few numerical experiments are conducted to confirm the theoretical conclusions.

Provably Positive Central Discontinuous Galerkin Schemes via Geometric Quasilinearization for Ideal MHD Equations

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124–2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov’s modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta.

Optimal error estimates to smooth solutions of the central discontinuous Galerkin methods for nonlinear scalar conservation laws

In this paper, we study the error estimates to sufficiently smooth solutions of the nonlinear scalar conservation laws for the semi-discrete central discontinuous Galerkin (DG) finite element methods on uniform Cartesian meshes. A general approach with an explicitly checkable condition is established for the proof of optimal L2 error estimates of the semi-discrete CDG schemes, and this condition is checked to be valid in one and two dimensions for polynomials of degree up to k = 8. Numerical experiments are given to verify the theoretical results.

Positivity-preserving well-balanced central discontinuous Galerkin schemes for the Euler equations under gravitational fields

This paper designs and analyzes positivity-preserving well-balanced (WB) central discontinuous Galerkin (CDG) schemes for the Euler equations with gravity. A distinctive feature of these schemes is that they not only are WB for a general known stationary hydrostatic solution, but also can preserve the positivity of the fluid density and pressure. The standard CDG method does not possess this feature, while directly applying some existing WB techniques to the CDG framework may not accommodate the positivity and keep other important properties at the same time. In order to obtain the WB and positivity-preserving properties simultaneously while also maintaining the conservativeness and stability of the schemes, a novel spatial discretization is devised in the CDG framework based on suitable modifications to the numerical dissipation term and the source term approximation. The modifications are based on a crucial projection operator for the stationary hydrostatic solution, which is proposed for the first time in this work. This novel projection has the same order of accuracy as the standard L2-projection, can be explicitly calculated, and is easy to implement without solving any optimization problems. More importantly, it ensures that the projected stationary solution has the same cell averages on both the primal and dual meshes, which is a key to achieve the desired properties of our schemes. Based on some convex decomposition techniques, rigorous positivity-preserving analyses for the resulting WB CDG schemes are carried out. Several one- and two-dimensional numerical examples are performed to illustrate the desired properties of these schemes, including the high-order accuracy, the WB property, the robustness for simulations involving the low pressure or density, high resolution for the discontinuous solutions and the small perturbations around the equilibrium state.

Error analysis of a fully discrete discontinuous Galerkin alternating direction implicit discretization of a class of linear wave-type problems

This paper is concerned with the rigorous error analysis of a fully discrete scheme obtained by using a central fluxes discontinuous Galerkin (dG) method in space and the Peaceman–Rachford splitting scheme in time. We apply the scheme to a general class of wave-type problems and show that the resulting approximations as well as discrete derivatives thereof satisfy error bounds of the order of the polynomial degree used in the dG discretization and order two in time. In particular, the class of problems considered includes, e.g., the advection equation, the acoustic wave equation, and the Maxwell equations for which a very efficient implementation is possible via an alternating direction implicit splitting.

Central discontinuous Galerkin methods on overlapping meshes for wave equations

In this paper, we study the central discontinuous Galerkin (DG) method on overlapping meshes for second order wave equations. We consider the first order hyperbolic system, which is equivalent to the second order scalar equation, and construct the corresponding central DG scheme. We then provide the stability analysis and the optimal error estimates for the proposed central DG scheme for one- and multi-dimensional cases with piecewise Pk elements. The optimal error estimates are valid for uniform Cartesian meshes and polynomials of arbitrary degree k ≥ 0. In particular, we adopt the techniques in Liu et al. (SIAM J. Numer. Anal. 56 (2018) 520–541; ESAIM: M2AN 54 (2020) 705–726) and obtain the local projection that is crucial in deriving the optimal order of convergence. The construction of the projection here is more challenging since the unknowns are highly coupled in the proposed scheme. Dispersion analysis is performed on the proposed scheme for one dimensional problems, indicating that the numerical solution with P1 elements reaches its minimum with a suitable parameter in the dissipation term. Several numerical examples including accuracy tests and long time simulation are presented to validate the theoretical results.

Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

Central Discontinuous Galerkin Methods for the Generalized Korteweg-de Vries Equation

A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property

In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method, which possesses an important feature of the needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features, which are highly desirable in the simulation of the shallow water wave, will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests.

Maximum-Principle-Satisfying and Positivity-Preserving High Order Central DG Methods on Unstructured Overlapping Meshes for Two-Dimensional Hyperbolic Conservation Laws

In this paper, we first present a family of high order central discontinuous Galerkin methods defined on unstructured overlapping meshes for the two-dimensional conservation laws. The primal mesh is a triangulation of the computational domain, while the dual mesh is a quadrangular partition which is formed by connecting an interior point and the three vertexes of each triangle on the primal mesh. We prove the \(L^2\) stability of the present method for linear equation. Then we design and analyze high order maximum-principle-satisfying central discontinuous Galerkin methods for two-dimensional scalar conservation law, and high order positivity-preserving central discontinuous Galerkin methods for two-dimensional compressible Euler systems. The performance of the proposed methods is finally demonstrated through a set of numerical experiments.

Sparse Grid Central Discontinuous Galerkin Method for Linear Hyperbolic Systems in High Dimensions

In this paper, we develop sparse grid central discontinuous Galerkin (CDG) scheme for linear hyperbolic systems with variable coefficients in high dimensions. The scheme combines the CDG framework with the sparse grid approach, with the aim of breaking the curse of dimensionality. A new hierarchical representation of piecewise polynomials on the dual mesh is introduced and analyzed, resulting in a sparse finite element space that can be used for non-periodic problems. Theoretical results, such as $L^2$ stability and error estimates are obtained for scalar problems. CFL conditions are studied numerically comparing discontinuous Galerkin (DG), CDG, sparse grid DG and sparse grid CDG methods. Numerical results including scalar linear equations, acoustic and elastic waves are provided.

A modified central discontinuous Galerkin method with positivity-preserving and well-balanced properties for the one-dimensional nonlinear shallow water equations

In this paper, we present a modified central discontinuous Galerkin method for solving the nonlinear shallow water equations over variable bottom topography in one-dimensional space. The proposed method balances exactly the flux gradient and source term in the still-water stationary case by adding a correction term to the base scheme and ensures non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter, meanwhile reduces the computational cost considerably compared with the standard central discontinuous Galerkin method by using L 2 projection to obtain an approximation solution. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.

Optimal Error Estimates of the Semidiscrete Central Discontinuous Galerkin Methods for Linear Hyperbolic Equations

We analyze the central discontinuous Galerkin method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor-product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.

Runge-Kutta Central Discontinuous Galerkin Methods for the Special Relativistic Hydrodynamics

AbstractThis paper develops Runge-Kutta PK-based central discontinuous Galerkin (CDG) methods with WENO limiter for the one- and two-dimensional special relativistic hydrodynamical (RHD) equations, K = 1,2,3. Different from the non-central DG methods, the Runge-Kutta CDG methods have to find two approximate solutions defined on mutually dual meshes. For each mesh, the CDG approximate solutions on its dual mesh are used to calculate the flux values in the cell and on the cell boundary so that the approximate solutions on mutually dual meshes are coupled with each other, and the use of numerical flux will be avoided. The WENO limiter is adaptively implemented via two steps: the “troubled” cells are first identified by using a modified TVB minmod function, and then the WENO technique is used to locally reconstruct new polynomials of degree (2K+1) replacing the CDG solutions inside the “troubled” cells by the cell average values of the CDG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells. Because the WENO limiter is only employed for finite “troubled” cells, the computational cost can be as little as possible. The accuracy of the CDG without the numerical dissipation is analyzed and calculation of the flux integrals over the cells is also addressed. Several test problems in one and two dimensions are solved by using our Runge-Kutta CDG methods with WENO limiter. The computations demonstrate that our methods are stable, accurate, and robust in solving complex RHD problems.

A reconstructed central discontinuous Galerkin method for conservation laws

Abstract Central discontinuous Galerkin (CDG) methods are a family of high order numerical methods, which evolve two sets of numerical solutions defined on overlapping meshes and do not employ the numerical fluxes at element interfaces as in discontinuous Galerkin methods. However, evolving two sets of numerical solutions makes the CDG method relatively time-consuming. In this paper, we present a reconstructed central discontinuous Galerkin (RCDG) method for conservation laws. In this scheme, we reconstruct an approximate solution by projecting the numerical solution defined on the primal mesh into the approximate space defined on the dual mesh in the L 2 sense. As a consequence, the reconstructed approximate solution plays a role as an alternative for the numerical solution defined on the dual mesh in the CDG method. Numerical examples demonstrate that the RCDG method is not only simpler and easier to implement than the CDG method but also reduces the computational cost considerably without sacrificing the high order accuracy of the CDG method.

Runge–Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics

This paper develops PK-based non-central and central Runge–Kutta discontinuous Galerkin (DG) methods with WENO limiter for the one- and two-dimensional special relativistic magnetohydrodynamical (RMHD) equations, K=1,2,3. The non-central DG methods are locally divergence-free, while the central DG are “exactly” divergence-free but have to find two approximate solutions defined on mutually dual meshes. The adaptive WENO limiter first identifies the “troubled” cells by using a modified TVB minmod function, and then uses the WENO technique and the cell average values of the DG solutions in the neighboring cells as well as the original cell averages of the “troubled” cells to locally reconstruct a new polynomial of degree (2K+1) inside the “troubled” cells instead of the DG solution. The WENO limiting procedure does not destroy the locally or “exactly” divergence-free property of magnetic field and is only employed for finite “troubled” cells so that the computational cost can be as little as possible. Several test problems in one and two dimensions are solved by using our non-central and central Runge–Kutta DG methods with WENO limiter. The numerical results demonstrate that our methods are stable, accurate, and robust in resolving complex wave structures.

非线性浅水波方程的保持平衡的中心间断伽辽金法

提出了一个求解非线性浅水波方程的高阶的保持平衡的中心间断伽辽金法，并证明了该方法的保持平衡性。与传统的中心间断伽辽金法相比，该方法能够精确地保持非线性浅水波方程的静水稳定解，因而消除了数值震荡。数值算例验证了方法的精度和可靠性。 A high order well-balanced central discontinuous Galerkin method is developed for solving the nonlinear shallow water equations and the well-balanced property is proved. Compared to the standard central discontinuous Galerkin method, the present method can maintain the still water stationary solutions of the nonlinear shallow water equations, and thus remove the numerical os-cillations. Numerical examples are presented to show the accuracy and reliability of the proposed method.

PHYSICAL-CONSTRAINT-PRESERVING CENTRAL DISCONTINUOUS GALERKIN METHODS FOR SPECIAL RELATIVISTIC HYDRODYNAMICS WITH A GENERAL EQUATION OF STATE

ABSTRACT The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax–Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L 1-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc.

A Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Method for the Nonlinear Shallow Water Equations

In this paper, we consider the development of central discontinuous Galerkin methods for solving the nonlinear shallow water equations over variable bottom topography in one and two dimensions. A reliable numerical scheme for these equations should preserve still-water stationary solutions and maintain the non-negativity of the water depth. We propose a high-order technique which exactly balances the flux gradients and source terms in the still-water stationary case by adding correction terms to the base scheme, meanwhile ensures the non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.

New central and central discontinuous Galerkin schemes on overlapping cells of unstructured grids for solving ideal magnetohydrodynamic equations with globally divergence-free magnetic field

New schemes are developed on triangular grids for solving ideal magnetohydrodynamic equations while preserving globally divergence-free magnetic field. These schemes incorporate the constrained transport (CT) scheme of Evans and Hawley [34] with central schemes and central discontinuous Galerkin methods on overlapping cells which have no need for solving Riemann problems across cell edges where there are discontinuities of the numerical solution. These schemes are formally second-order accurate with major development on the reconstruction of globally divergence-free magnetic field on polygonal dual mesh. Moreover, the computational cost is reduced by solving the complete set of governing equations on the primal grid while only solving the magnetic induction equation on the polygonal dual mesh. Various numerical experiments are provided to validate the new schemes.