Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – Speed comparisons with Runge–Kutta methods

Balsara, Dinshaw S.; Meyer, Chad D.; Dumbser, Michael; Du, Huijing; Xu, Zhiliang

doi:10.1016/j.jcp.2012.04.051

Cited by 119 publications

(157 citation statements)

References 69 publications

(212 reference statements)

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“…The AMR technology is used to refine and recoarsen the computational grid according to physical features that one wants to follow, while the subgrid limiter is only used to cope with shock waves or other discontinuities that require limiting of the DG scheme. In the following we provide the necessary minimum details for each of the above items, while addressing the reader to [41,51,67,58,11,52,53] for an exhaustive discussion of the subtleties that may be involved.…”

Section: Mathematical Framework: An Overviewmentioning

confidence: 99%

“…At the heart of the ADER approach, either in the original version proposed in [113,115] or in the later version proposed in [43,41,11], that we also follow in this paper, there is the solution of the generalized or derivative Riemann problem. This requires a time evolution of known spatial derivatives of the polynomials approximating the solution at time t n and is in our case performed locally for each cell and independently from the neighbor cells.…”

Section: The Local Space-time Predictormentioning

confidence: 99%

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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: Mathematical Framework: An Overviewmentioning

confidence: 99%

Section: The Local Space-time Predictormentioning

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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“…In most of the approaches mentioned so far the evolution in time is performed through the method of lines, resulting in multistep Runge-Kutta schemes, either explicit or implicit. A valuable alternative is provided by ADER schemes, which were introduced by Titarev & Toro (2005); and became popular after the modern reformulation by ; ; Balsara et al (2013). In a nutshell, ADER schemes are high order numerical schemes with a single step for the time update and they have been already applied to the equations of relativistic MHD, both in the ideal case and in the resistive case (Dumbser & Zanotti 2009).…”

Section: Introductionmentioning

confidence: 99%

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

Zanotti

Fambri

Dumbser

2015

Mon. Not. R. Astron. Soc.

114

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We present a new numerical tool for solving the special relativistic ideal MHD equations that is based on the combination of the following three key features: (i) a one-step ADER discontinuous Galerkin (DG) scheme that allows for an arbitrary order of accuracy in both space and time, (ii) an a posteriori subcell finite volume limiter that is activated to avoid spurious oscillations at discontinuities without destroying the natural subcell resolution capabilities of the DG finite element framework and finally (iii) a space-time adaptive mesh refinement (AMR) framework with time-accurate local time-stepping.The divergence-free character of the magnetic field is instead taken into account through the so-called "divergence-cleaning" approach. The convergence of the new scheme is verified up to 5 th order in space and time and the results for a set of significant numerical tests including shock tube problems, the RMHD rotor and blast wave problems, as well as the Orszag-Tang vortex system are shown. We also consider a simple case of the relativistic Kelvin-Helmholtz instability with a magnetic field, emphasizing the potential of the new method for studying turbulent RMHD flows. We discuss the advantages of our new approach when the equations of relativistic MHD need to be solved with high accuracy within various astrophysical systems.

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“…Further generalisations were put forward by Dumbser et al [12], setting ADER-FV and ADER-DG in a generalised framework. The ADER approach has undergone numerous extensions and applications, examples include [2,21,23,22,35,36,37,40,41,12,5,6,10,11,3,9,7,1,28,29,31,13,14,30]. A succinct review of the ADER approach, in the frame of the finite volume method, is presented here, in terms of a one-dimensional system of hyperbolic balance laws…”

Section: The Ader Approachmentioning

confidence: 99%

A novel numerical flux for the 3D Euler equations with general equation of state

Toro

Castro

Lee

2015

Journal of Computational Physics

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Please cite this article in press as: E. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractHere we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on general meshes; (ii) we use a general equation of state; and (iii) we achieve high order of accuracy in both space and time through application of the semi-discrete ADER methodology on general meshes. The resulting methods are systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems. Formal high accuracy is assessed through convergence rates studies for schemes of up to 4th order of accuracy in both space and time on unstructured meshes.

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Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – Speed comparisons with Runge–Kutta methods

Cited by 119 publications

References 69 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

Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement

A novel numerical flux for the 3D Euler equations with general equation of state

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