We present a discontinuous Galerkin internal-penalty scheme that is applicable to a large class of linear and non-linear elliptic partial differential equations. The scheme constitutes the foundation of the elliptic solver for the SpECTRE numerical relativity code. As such it can accommodate (but is not limited to) elliptic problems in linear elasticity, general relativity and hydrodynamics, including problems formulated on a curved manifold. We provide practical instructions that make the scheme functional in a production code, such as instructions for imposing a range of boundary conditions, for implementing the scheme on curved and non-conforming meshes and for ensuring the scheme is compact and symmetric so it may be solved more efficiently. We report on the accuracy of the scheme for a suite of numerical test problems.
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This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new approach has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.
We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.
We analyze the spatial structure of asymptotics of a solution to a singularly perturbed system of mass transfer equations. The leading term of the asymptotics is described by a parabolic equation with possibly degenerate spatial part. We prove a theorem that establishes a relationship between the degree of degeneracy and the numbers of equations in the system and spatial variables in some particular cases. The work hardly depends on the calculation of the eigenvalues of matrices that determine the spatial structure of the asymptotics by the means of computer algebra system Wolfram Mathematica. We put forward a hypothesis on the existence of the found connection for an arbitrary number of equations and spatial variables.
We consider an iterative procedure to solve quasilinear elliptic systems with $p$-growth. The scheme was first considered by Koshelev in the quadratic case $p=2$. We present numerical applications as well as applications to higher regularity properties.
This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.
The dispersion error is often the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of $\alpha$-dispersion-relation-preserving schemes. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which preserve the dispersion-relation of the continuous problem uniformly to an $\alpha \%$-error tolerance. We give a general framework to design provably stable finite difference operators that preserve the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ($\pi$-mode) present on any equidistant grid at a tolerance of $5\%$ error. This significantly improves on the current standard that have a tolerance of $100 \%$ error.
One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.
Sakurai et al. (J Comput Phys, 2019) presented a flux-based volume penalization (VP) approach for imposing inhomogeneous Neumann boundary conditions on embedded interfaces. The flux-based VP method modifies the diffusion coefficient of the original elliptic (Poisson) equation and uses a flux-forcing function as a source term in the equation to impose the Neumann boundary conditions. As such, the flux-based VP method can be easily incorporated into existing fictitious domain codes. Sakurai et al. relied on an analytical construction of flux-forcing functions, which limits the practicality of the approach. Because of the analytical approach taken in the prior work, only (spatially) constant flux values on simple interfaces were considered. In this paper, we present a numerical technique for constructing flux-forcing functions for arbitrarily complex boundaries. The imposed flux values are also allowed to vary spatially in our approach. Furthermore, the flux-based VP method is extended to include (spatially varying) Robin boundary conditions, which makes the flux-based VP method even more general. We consider several two- and three-dimensional test examples to access the spatial accuracy of the numerical solutions. The method is also used to simulate flux-driven thermal convection in a concentric annular domain. We formally derive the flux-based volume penalized Poisson equation satisfying Neumann/Robin boundary condition in strong form; such a derivation was not presented in Sakurai et al., where the equation first appeared for the Neumann problem. The derivation reveals that the flux-based VP approach relies on a surface delta function to impose inhomogeneous Neumann/Robin boundary conditions. However, explicit construction of the delta function is not necessary for the flux-based VP method, which makes it different from other diffuse domain equations presented in the literature.
In this article, we develop and analyse a new spectral method to solve the semi-classical Schr\"odinger equation based on the Gaussian wave-packet transform (GWPT) and Hagedorn's semi-classical wave-packets (HWP). The GWPT equivalently recasts the highly oscillatory wave equation as a much less oscillatory one (the $w$ equation) coupled with a set of ordinary differential equations governing the dynamics of the so-called GWPT parameters. The Hamiltonian of the $ w $ equation consists of a quadratic part and a small non-quadratic perturbation, which is of order $ \mathcal{O}(\sqrt{\varepsilon }) $, where $ \varepsilon\ll 1 $ is the rescaled Planck's constant. By expanding the solution of the $ w $ equation as a superposition of Hagedorn's wave-packets, we construct a spectral method while the $ \mathcal{O}(\sqrt{\varepsilon}) $ perturbation part is treated by the Galerkin approximation. This numerical implementation of the GWPT avoids imposing artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional cases, we establish how the error of solving the semi-classical Schr\"odinger equation with the GWPT is determined by the errors of solving the $ w $ equation and the GWPT parameters. We prove that this scheme has the spectral convergence with respect to the number of Hagedorn's wave-packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method.
Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, BBM-BBM system, Bona-Smith system etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.
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