Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes

Zhu, Jun; Zhong, Xinghui; Shu, Chi‐Wang; Qiu, Jianxian

doi:10.1016/j.jcp.2013.04.012

Cited by 150 publications

(64 citation statements)

References 27 publications

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“…In the discontinuous Galerkin approach, on the other hand, even if in principle no spatial reconstruction is needed, in practice it is necessary to introduce some sort of limiters to avoid oscillations in the presence of discontinuities. Among the most relevant limiters proposed so far we mention the use of artificial viscosity [66,98,26,56,38,55], of spectral filtering [103], of (H) WENO limiting procedures [102,101,77,4,78,79,68], and of slope and moment limiting [30,88,104,1,125,37]. In [53] we have recently proposed a totally different and alternative solution to this longstanding problem, which relies on a new a posteriori sub-cell finite volume limiting approach.…”

Section: Introductionmentioning

confidence: 99%

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

et al. 2015

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A posteriori sub-cell finite volume limiter MOOD paradigm High order space-time adaptive mesh refinement (AMR) ADER-DG and ADER-WENO finite volume schemes Hyperbolic conservation laws a b s t r a c t In this paper we present a novel arbitrary high order accurate discontinuous Galerkin (DG) finite element method on space-time adaptive Cartesian meshes (AMR) for hyperbolic conservation laws in multiple space dimensions, using a high order a posteriori sub-cell ADER-WENO finite volume limiter. Notoriously, the original DG method produces strong oscillations in the presence of discontinuous solutions and several types of limiters have been introduced over the years to cope with this problem. Following the innovative idea recently proposed in [53], the discrete solution within the troubled cells is recomputed by scattering the DG polynomial at the previous time step onto a suitable number of sub-cells along each direction. Relying on the robustness of classical finite volume WENO schemes, the sub-cell averages are recomputed and then gathered back into the DG polynomials over the main grid. In this paper this approach is implemented for the first time within a space-time adaptive AMR framework in two and three space dimensions, after assuring the proper averaging and projection between sub-cells that belong to different levels of refinement. The combination of the sub-cell resolution with the advantages of AMR allows for an unprecedented ability in resolving even the finest details in the dynamics of the fluid. The spectacular resolution properties of the new scheme have been shown through a wide number of test cases performed in two and in three space dimensions, both for the Euler equations of compressible gas dynamics and for the magnetohydrodynamics (MHD) equations.

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Section: Introductionmentioning

confidence: 99%

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

et al. 2015

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“…Numerical results in one and two dimensions are provided to show that this WENO limiting procedure can simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. In future work we will extend the WENO limiter to CPR framework on unstructured meshes along the lines of [47].…”

Section: Resultsmentioning

confidence: 99%

“…This simple WENO limiter attempts to reconstruct the entire polynomial in one shot, using the information only from the target cell and its immediate neighbors. Extension to 2D triangulations is studied in [47].…”

Section: Introductionmentioning

confidence: 99%

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework

Shu

Zhang

2015

Applied Numerical Mathematics

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“…Generally, the discontinuous Galerkin spatial discretization is paired with a multi-step RungeKutta numerical ODE solver for temporal discretization [27,28,32,33,22,34]. RKDG is conceptually simple and easy to implement, but it has drawbacks.…”

Section: Advanced Discontinuous Galerkin Schemesmentioning

confidence: 99%

A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

Reddell¹

2016

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A continuum kinetic model for plasma based on the Vlasov-Maxwell system for multiple particle species is developed. Consideration is added for boundary conditions in a truncated velocity domain and supporting wall interactions. A scheme to scale the velocity domain for multiple particle species with different temperatures and particle mass while sharing one computational mesh is described. A method for assessing the degree to which the kinetic solution differs from a Maxwell-Boltzmann distribution is introduced and tested on a thoroughly studied test case.The discontinuous Galerkin numerical method is extended for efficient solution of hyperbolic conservation laws in five or more particle phase-space dimensions using tensor-product hypercube elements with arbitrary polynomial order. A scheme for velocity moment integration is integrated as required for coupling between the plasma species and electromagnetic waves.A new high performance simulation code WARPM is developed to efficiently implement the model and numerical method on emerging many-core supercomputing architectures.WARPM uses the OpenCL programming model for computational kernels and task parallelism to overlap computation with communication. WARPM single-node performance and parallel scaling efficiency are analyzed with bottlenecks identified guiding future directions for the implementation.The plasma modeling capability is validated against physical problems with analytic solutions and well established benchmark problems.

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Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes

Cited by 150 publications

References 27 publications

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework

A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

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