Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

Dumbser, Michael; Käser, Martin; Титарев, В. А.; Toro, Eleuterio F.

doi:10.1016/j.jcp.2007.04.004

Cited by 316 publications

(357 citation statements)

References 60 publications

(119 reference statements)

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“…However according to reference [16] the MOOD method is competitive when compared to truly unstructured methods of the same order. Table 5 Memory storage and total number of iterations for the Double Mach problem according to the different configurations with the MOOD method.…”

Section: Variablesmentioning

confidence: 99%

“…Since this date, a large literature was dedicated to the limited reconstruction methods for structured and unstructured meshes. Several strategies became very popular due to their intrinsic simplicity such that the MUSCL method [3,6,7,20,26,32] or their efficiency to achieve very high-order accuracy such that the ENO/WENO method [1,21,29,30,36,37,46,18,19,35,49,41], the Discontinuous Galerkin method [9][10][11][12], the ADER method [14,39,40,15,16], the Residual Distribution Scheme [2,13,33] and the spectral method [17,44,45].…”

Section: Introductionmentioning

confidence: 99%

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Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

2012

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This paper extends the MOOD method proposed by the authors in ["A high-order finite volume method for hyperbolic systems: Multi-dimensional Optimal Order Detection (MOOD)", J. Comput. Phys. 230, pp 4028-4050, (2011)], along two complementary axes: extension to very high-order polynomial reconstruction on nonconformal unstructured meshes and new Detection Criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the Discrete Maximum Principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient.

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Section: Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

2012

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“…This is facilitated by our choice of basis functions with zero average, so that it is sufficient to solve the equations (9) for k ≥ 2 as an unconstrained least squares problem. Finally, we point out that the stencil S i employed in this work is strictly smaller than the stencils employed by [DK07,DKTT07a] for the same level of accuracy. S i is the same as the stencil of the CWENO3 reconstruction of [SCR15].…”

Section: Polynomial Reconstruction Operatormentioning

confidence: 93%

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

Semplice

Loubère

2018

Journal of Computational Physics

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To cite this version:Matteo Semplice, Raphaël Loubère. Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions. AbstractIn this paper we propose a third order accurate finite volume scheme based on polynomial reconstruction along with a posteriori limiting within an Adaptive-Mesh-Refinement (AMR) simulation code for hydrodynamics equations in 2D. The a posteriori limiting is based on the detection of problematic cells on a so-called candidate solution computed at each stage of a third order Runge-Kutta scheme. Such detection may include different properties, derived from physics, such as positivity, from numerics, such as a nonoscillatory behavior, or from computer requirements such as the absence of NaN's. Troubled cell values are discarded and re-computed starting again from the previous time-step using this time a more dissipating scheme but only locally to these cells. By decrementing the degree of the polynomial reconstructions from 2 to 0 we switch from a third-order to a first-order accurate scheme. Ultimately some troubled cells may be updated with a first order accurate scheme for instance close to steep gradients. The entropy indicator sensor is used to refine/coarsen the mesh. This sensor is also employed in an a posteriori manner because if some refinement is needed at the end of a time step, then the current time-step is recomputed but only locally with such refined mesh. We show on a large set of numerical tests that this a posteriori limiting procedure coupled with the entropy-based AMR technology not only can maintain optimal accuracy on smooth flows but also stability on discontinuous profiles such as shock waves, contacts, interfaces, etc. Moreover numerical evidences show that this approach is comparable in terms of accuracy and cost to a more classical CWENO approach within the same AMR context.

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“…Recently, Dumbser [16] adopted the Arbitrary high order schemes using DERivatives (ADER) WENO finite volume schemes to the Baer-Nunziato model of compressible multi-fluid flows. However, this WENO reconstruction for multi-fluid flows in literature is performed on the structured grid, and their extension to unstructured grid is not trivial [17][18][19][20]. The main reason is that the expansion of stencil, which is quite straightforward in structured grid, becomes difficult because of the complicated topology in unstructured grid.…”

Section: Introductionmentioning

confidence: 99%

An improved unstructured WENO method for compressible multi‐fluids flow

Zheng

Zhao

2016

Numerical Methods in Fluids

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SUMMARYAn improved high-order accurate WENO finite volume method based on unstructured grids for compressible multi-fluids flow is proposed in this paper. The third-order accuracy WENO finite volume method based on triangle cell is used to discretize the governing equations. To have higher order of accuracy, the P1 polynomial is reconstructed firstly. After that, the P2 polynomial is reconstructed from the combination of the P1. The reconstructed coefficients are calculated by analytical form of inverse matrix rather than the numerical inversion. This greatly improved the efficiency and the robustness. Four examples are presented to examine this algorithm. Numerical results show that there is no spurious oscillation of velocity and pressure across the interface and high-order accurate result can be achieved.

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Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems

Cited by 316 publications

References 60 publications

Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

Improved detection criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

Adaptive-Mesh-Refinement for hyperbolic systems of conservation laws based on a posteriori stabilized high order polynomial reconstructions

An improved unstructured WENO method for compressible multi‐fluids flow

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