High Order Cut Discontinuous Galerkin Methods for Hyperbolic Conservation Laws in One Space Dimension

Fu, Pei; Kreiss, Gunilla

doi:10.1137/20m1349060

Cited by 12 publications

(12 citation statements)

References 25 publications

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“…Examining (12), we observe that for α ≥ ν there holds η k1 = 0 and therefore the stabilization J h vanishes. This is intended as in this case the standard CFL condition on cell I k1 is satisfied and we do not have a small cell problem.…”

Section: Choice Of η K1mentioning

confidence: 91%

“…While there are many different approaches for stabilizing discretizations for elliptic and parabolic problems on cut cell meshes (for an overview see, e.g., [5]) the research for hyperbolic problems is still at the beginning but with a lot of current activity. Some very recent work [12,16,32] is based on applying the ghost penalty stabilization [6], which is a well-known approach for elliptic equations, to hyperbolic problems. Out of these contributions, only Fu and Kreiss [12] address the small cell problem for first-order hyperbolic problems by developing a stabilization for the solution of scalar conservation laws in one dimension.…”

Section: Introductionmentioning

confidence: 99%

“…Some very recent work [12,16,32] is based on applying the ghost penalty stabilization [6], which is a well-known approach for elliptic equations, to hyperbolic problems. Out of these contributions, only Fu and Kreiss [12] address the small cell problem for first-order hyperbolic problems by developing a stabilization for the solution of scalar conservation laws in one dimension. A different approach to overcoming the small cell problem was taken by Giuliani [13] who extends the state redistribution scheme to the DG setting.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

May¹,

Streitbürger²

2021

Preprint

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In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space and an explicit strong stability preserving Runge-Kutta scheme in time. When applied on a cut cell mesh with a time step length that is appropriate for the size of the larger background cells, one encounters stability issues. The DoD stabilization consists of penalty terms that are designed to address these problems by redistributing mass between the inflow and outflow neighbors of small cut cells in a physical way. For piecewise constant polynomials in space and explicit Euler in time, the stabilized scheme is monotone for scalar problems. For higher polynomial degrees p, our numerical experiments show convergence orders of p + 1 for smooth flow and robust behavior in the presence of shocks.

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Section: Choice Of η K1mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

May¹,

Streitbürger²

2021

Preprint

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“…For hyperbolic problems, CutFEM based on discontinuous piecewise polynomial spaces and ghost penalty stabilization has been developed, e.g. see [12] where a time independent linear advection-reaction problem is considered and see [10] for time dependent nonlinear conservation laws. We also refer to the recent work [8] where a DG method for time dependent linear advection problems is developed with a stabilization of small elements that is designed to restore proper domains of dependence.…”

Section: Introductionmentioning

confidence: 99%

“…The first main result is an extension of the family of high order CutFEM with ghost penalty stabilization in [10] to problems with stationary interfaces. In the new method the solution is built from separate solutions on the two sides of the interface, and coupled through the interface condition, which is imposed weakly through penalties in the weak form.…”

Section: Introductionmentioning

confidence: 99%

High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface

Pei¹,

Frachon²,

Kreiss³

et al. 2022

Preprint

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We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

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Accuracy considerations of mixed explicit implicit schemes for embedded boundary meshes

May

Berger

Fabian

2019

Proc Appl Math and Mech

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We consider a mixed explicit implicit finite volume scheme for solving the advection equation on a Cartesian cut cell mesh: the scheme couples an implicit time stepping scheme on cut cells with an explicit time stepping scheme on cells away from the cut cells by means of flux bounding. We examine the accuracy of the resulting mixed scheme to see if a second-order explicit and a second-order implicit scheme result in a second-order mixed scheme.

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High Order Cut Discontinuous Galerkin Methods for Hyperbolic Conservation Laws in One Space Dimension

Cited by 12 publications

References 25 publications

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

DoD Stabilization for non-linear hyperbolic conservation laws on cut cell meshes in one dimension

High order discontinuous cut finite element methods for linear hyperbolic conservation laws with an interface

Accuracy considerations of mixed explicit implicit schemes for embedded boundary meshes

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