An h-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws

Zhu, Hongqiang; Qiu, Jianxian

doi:10.1007/s10444-012-9287-7

Cited by 24 publications

(21 citation statements)

References 20 publications

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“…In order to retain the approximation accuracy and the local conservation character, a L 2 projection approach is used to obtain the new generated grid's degrees of freedom [12,28]. Suppose U h is already known on the mesh T h (t n ), and we need to determine the degrees of freedom U (j)…”

Section: H-adaptive Rkdg Methodsmentioning

confidence: 99%

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Liu

Qiu

Goman

et al. 2016

Numer. Math. Theory Methods Appl.

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Abstract. In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

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Section: H-adaptive Rkdg Methodsmentioning

confidence: 99%

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Liu

Qiu

Goman

et al. 2016

Numer. Math. Theory Methods Appl.

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“…The time integration of Eq. (17) can be performed through Runge-Kutta schemes, leading to RKDG schemes (Cockburn & Shu 1991Cockburn et al , 1990Cockburn & Shu 1998;H.Zhu & J.Qiu 2013) but it can also be obtained through the ADER philosophy, see (Dumbser & Munz 2006;). More precisely, we want to devise a one-step time integration scheme for (17), while preserving high order of accuracy both in space and in time.…”

Section: The Discontinuous Galerkin Schemementioning

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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“…For nonlinear hyperbolic problems that developing discontinuities, meshes near the discontinuities should be refined in order to catch sharp transitions of discontinuities and coarse meshes should be used in other regions to save the computational cost. Several troubled-cell indicators are used as adaptive indicators to detect the discontinuous regions with success in [38,39]. However, the VP solutions are known to remain smooth given smooth initial conditions [29], yet develop filamentation solution structures with large gradients in the phase space over time.…”

Section: Refinement Criteriamentioning

confidence: 99%

“…Each cell in the initial partition of the computational domain is considered as the root of a tree. Following our previous work in [39], we adopt a simple way to perform local mesh refinement and coarsening. More specifically, mesh refinement is achieved by dividing a cell into four new cells (children) of equal size (see the right sketch in Fig.…”

Section: Grid and Data Structurementioning

confidence: 99%

“…Moreover, the DG framework offers great convenience in data projection and data prolongation among different levels of mesh refinement, while preserving the mass conservation. Existing h-adaptive RKDG schemes include the ones developed by Flaherty et al [4,13,18,27], Zhu and Qiu [38,39] and Dedner et al [12] for nonlinear time-dependent hyperbolic conservation laws. We also refer to Hartmann and Houston [20], where duality techniques were used for designing the adaptive strategy.…”

Section: Introductionmentioning

confidence: 99%

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An h-Adaptive RKDG Method for the Vlasov–Poisson System

Zhu

Qiu

2016

J Sci Comput

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In this paper, we propose a new h-adaptive indicator for the Runge-Kutta discontinuous Galerkin (RKDG) scheme in simulations of the Vlasov-Poisson (VP) system. This adaptive indicator, tailored for the VP system, is based on the principle that each cell assumes solution variations as equally as possible. Under the framework of the RKDG method, such adaptive indicator is particularly simple and cheap for the computation. Its effectiveness is demonstrated by extensive numerical tests. The detailed adaptive algorithm as well as some important implementation issues, including the grid and data structure, adaptive criteria, data prolongation/projection and mesh projection, is presented.

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An h-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws

Cited by 24 publications

References 20 publications

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

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

An h-Adaptive RKDG Method for the Vlasov–Poisson System

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