A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

Dumbser, Michael; Balsara, Dinshaw S.; Toro, Eleuterio F.; Munz, Claus‐Dieter

doi:10.1016/j.jcp.2008.05.025

Cited by 572 publications

(680 citation statements)

References 79 publications

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“…The scheme can almost reach third-and fifth-order accuracy by using the third-and fifth-order WENO reconstruction, respectively. As analyzed by Dumbser et al [57], B x and B y have opposite signs on the left and right and on the top and bottom boundary, respectively. Hence, there is a jump in the magnetic field at the boundaries due to the periodic boundary conditions.…”

Section: Two-dimensional Iso-density Mhd Vortex Advectionmentioning

confidence: 89%

“…The 2D vortex advection problem proposed in [56] and lately considered by Dumbser et al [57] and Mignone at al. [58] is used to test the scheme's accuracy.…”

Section: Two-dimensional Iso-density Mhd Vortex Advectionmentioning

confidence: 99%

“…[58] is used to test the scheme's accuracy. The computational conditions are taken as the same used in [57,58]. The uniform meshes on the box ½À5; 5 Â ½À5; 5 with the following initial conditions are adopted.…”

Section: Two-dimensional Iso-density Mhd Vortex Advectionmentioning

confidence: 99%

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E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Shen

Zha

Huerta

2012

Journal of Computational Physics

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a b s t r a c tAn E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

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Section: Two-dimensional Iso-density Mhd Vortex Advectionmentioning

confidence: 89%

“…The 2D vortex advection problem proposed in [56] and lately considered by Dumbser et al [57] and Mignone at al. [58] is used to test the scheme's accuracy.…”

Section: Two-dimensional Iso-density Mhd Vortex Advectionmentioning

confidence: 99%

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E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Shen

Zha

Huerta

2012

Journal of Computational Physics

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“…(54) provides an explicit scheme which is stable under a CFL condition. The theoretical Courant number limits are CFL < 0.32 for a 2nd order scheme and, CFL < 0.17 for a 3rd order scheme, according to a linear stability analysis performed by Dumbser et al [16]. Such limits are in accordance with the usual ones for a DG scheme, depending on the degree of the polynomial representing the numerical solution.…”

Section: (45)mentioning

confidence: 53%

“…The procedure requires successive analytical differentiations and it is rather cumbersome. In order to avoid the analytical calculations involved with it, a method based on local DG predictors was proposed by Dumbser et al [16] for the case of a SCL with non stiff source terms and by Dumbser et al [3] in the case of a SCL with stiff source terms. The method was then extended to nonconservative systems by Dumbser et al [1].…”

Section: Numerical Schemementioning

confidence: 99%

Second order discontinuous Galerkin scheme for compound natural channels with movable bed. Applications for the computation of rating curves

Minatti¹,

Cicco

Solari

2016

Advances in Water Resources

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a b s t r a c tA new higher order 1D numerical scheme for the propagation of flood waves in compound channels with a movable bed is presented. The model equations are solved by means of an ADER Discontinuous Galerkin explicit scheme which can, in principle, reach any order of space-time accuracy. The higher order nature of the scheme allows the numerical coupling between flux and source terms appearing in the governing equations and, importantly, to handle moderately stiff and stiff source terms. Stiff source terms arise in the case of abrupt changes of river geometry such as in the case of hydraulic structures like bridges and weirs. Hydraulic interpretation of these conditions with 1D numerical modelling requires particular attention; for instance, a 1st order scheme might either lead to inaccurate solutions or impossibility to simulate these complex conditions. Validation is carried out with several test cases with the aim to check the scheme capability to deal with abrupt geometric changes and to capture the direction and celerity of propagation of bed and water surface disturbances. Validation is done also in a real case by using stage-discharge field measurements in the Ombrone river (Tuscany). The proposed scheme is further employed for the computation of flow rating curves in cross-sections just upstream of an abrupt narrowing, considering both fixed and movable bed conditions and different ratios of contraction for cross-section width. This problem is of particular relevance as, in common engineering practice, rating curves are derived from stage-measuring gauges installed on bridges with flow conditions that are likely to be influenced by local width narrowing. Results show that a higher order scheme is needed in order to deal with stiff source terms and reproduce realistic flow rating curves, unless a strong refinement of the computational grid is performed. This capability appears to be crucial for the computation of rating curves on coarse grids as it allows the modeling of abrupt contractions and jumps in bed bottom elevations, which often occur near cross-sections where stage measuring gauges are installed.

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Riemann Solvers in Aerospace Engineering

Toro

2010

Encyclopedia of Aerospace Engineering

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A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

Cited by 572 publications

References 79 publications

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

Second order discontinuous Galerkin scheme for compound natural channels with movable bed. Applications for the computation of rating curves

Riemann Solvers in Aerospace Engineering

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