Tracking discontinuities in hyperbolic conservation laws with spectral accuracy

Touil, Hatem; Hussaini, M. Y.; Sussman, Mark

doi:10.1016/j.jcp.2007.02.016

Cited by 9 publications

(10 citation statements)

References 32 publications

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“…At present, however, a mathematical understanding of the impact of filtering on the accuracy and the stability of Legendre pseudospectral methods is still lacking [56]. EDGES provides a number of different filter functions, but the most used are the exponential filter (66), which is only an approximate filter, but which is very popular due to its flexibility [55], and the ''erfc-log filter'' proposed by Boyd [58],…”

Section: Stabilization Techniquesmentioning

confidence: 99%

See 1 more Smart Citation

Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes

Radice

Rezzolla

2011

Phys. Rev. D

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We have developed the formalism necessary to employ the discontinuous-Galerkin approach in generalrelativistic hydrodynamics. The formalism is first presented in a general four-dimensional setting and then specialized to the case of spherical symmetry within a 3 þ 1 splitting of spacetime. As a direct application, we have constructed a one-dimensional code, EDGES, which has been used to assess the viability of these methods via a series of tests involving highly relativistic flows in strong gravity. Our results show that discontinuous-Galerkin methods are able not only to handle strong relativistic shock waves but, at the same time, to attain very high orders of accuracy and exponential convergence rates in smooth regions of the flow. Given these promising prospects and their affinity with a pseudospectral solution of the Einstein equations, discontinuous-Galerkin methods could represent a new paradigm for the accurate numerical modelling in relativistic astrophysics.

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Section: Stabilization Techniquesmentioning

confidence: 99%

“…[66], is that no interpolation is required in the switching from/to the discontinuous-Galerkin method. The coupling between the two methods is also very natural and is done through the Riemann solvers between the elements.…”

Section: Treatment Of Low-density Regionsmentioning

confidence: 99%

Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes

Radice

Rezzolla

2011

Phys. Rev. D

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“…It is well known that such problems may develop discontinuous solutions even if the initial values are smooth. In order to suppress the Gibbs-type oscillations caused by applying higher order numerical methods to discontinuous solutions, one can either introduce a dissipative term (e.g., [1]) or track the discontinuities explicitly (e.g., [21]) when the initial location of a discontinuity is known. In the former case, we rewrite (1) as,…”

Section: Problem Formulationmentioning

confidence: 99%

“…conservation laws. We refer to the article by Touil et al [21] and the references therein; they resorted to refining the mesh in discontinuous regions which introduces an inordinate number of additional degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

A Space-Time Discontinuous Galerkin Spectral Element Method for Nonlinear Hyperbolic Problems

Pei

Sussman

Hussaini

2018

Int. J. Comput. Methods

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A space-time discontinuous Galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions: (i) adding a space-time dependent artificial viscosity, and (ii) tracking the discontinuity with space-time spectral accuracy. A Picard iteration method is employed to solve nonlinear system of equations derived from the space-time DG spectral element discretization. Spectral accuracy in both space and time is demonstrated for the Burgers’ equation with a smooth solution. For tests with discontinuities, the present space-time method enables better accuracy at capturing the shock strength in the element containing shock when higher order polynomials in both space and time are used. The spectral accuracy of the shock speed and location is demonstrated for the solution of the inviscid Burgers’ equation obtained by the tracking method.

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“…De-Kang proposed a shock-tracking scheme based on a Godunov solver [7]; LeVeque and Shyue implemented shock tracking in two spatial dimensions within a finite volume scheme [8]. Finite element methods based on discrete spaces that admit discontinuities, including extended finite element, partition of unity and discontinuous spectral element schemes [9][10][11], have been used to implement tracking, usually in only one spatial dimension and with relatively simple shock systems.…”

Section: Introductionmentioning

confidence: 99%

Sub‐cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws

Palaniappan

Miller

Haber

2008

Numerical Methods in Fluids

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SUMMARYWe describe two approaches to resolving shocks and other discontinuities in spacetime discontinuous Galerkin (SDG) methods for nonlinear conservation laws. The first is an adaptation of the sub-cell shockcapturing technique, recently introduced by Persson and Peraire, to the special circumstances of SDG solutions constructed on causal spacetime grids. We restrict the stabilization operator to spacetime element interiors, thereby preserving the O(N ) computational complexity and the element-wise conservation properties of the basic SDG method, and use a special discontinuity indicator to limit the stabilization to elements traversed by discontinuous solution features. The method resolves discontinuities within individual spacetime elements having a sufficiently high-order basis. Numerical studies demonstrate the combination of sub-cell shock capturing with h-adaptive spacetime meshing that circumvents the projection errors inherent to purely spatial remeshing procedures.In a second method, we use adaptive spacetime meshing operations to track the trajectories of singular surfaces while maintaining the quality of the surrounding mesh. We present a series of feasibility studies where we track shocks and contact discontinuities in solutions to the inviscid Euler equations, including an example where the trajectories are not known a priori. The SDG-tracking method sharply resolves discontinuities without mesh refinement and requires very little stabilization.

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Tracking discontinuities in hyperbolic conservation laws with spectral accuracy

Cited by 9 publications

References 32 publications

Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes

Discontinuous Galerkin methods for general-relativistic hydrodynamics: Formulation and application to spherically symmetric spacetimes

A Space-Time Discontinuous Galerkin Spectral Element Method for Nonlinear Hyperbolic Problems

Sub‐cell shock capturing and spacetime discontinuity tracking for nonlinear conservation laws

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