Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

Boscheri, Walter; Loubère, Raphaël; Dumbser, Michael

doi:10.1016/j.jcp.2015.03.015

Cited by 60 publications

(52 citation statements)

References 117 publications

(237 reference statements)

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“…In the popular WENO technique has been used for the reconstruction procedure, that in the polynomial formulation presented in implies the use of seven (in 2D) or nine (in 3D) reconstruction stencils, each of them producing a reconstruction polynomial of the form () that has to be blended nonlinearly to obtain the final reconstruction polynomial

{boldw}_{h}^{n}

for each element

T_{i}^{n}

. On the contrary, the MOOD approach only needs one reconstruction stencil

{scriptS}_{i}

to derive one final reconstruction polynomial

{boldw}_{h}^{n}

, which is used as is, that is, unlimited . The limiting procedure is carried out a posteriori as follows: at the end of each timestep, the discrete candidate numerical solution

{boldQ}_{i}^{*}

, provided by the finite volume scheme, is checked against a set of so‐called detection criteria, and if the solution does not pass the checks, then the polynomial reconstruction degree M is locally decremented and the discrete solution

{boldQ}_{i}^{*}

is computed again with a lower polynomial degree.…”

Section: Methodsmentioning

confidence: 99%

“…High order of accuracy in time is then achieved using a local space‐time evolution of the reconstruction polynomials

{boldw}_{h}^{n}

, hence obtaining for each control volume

T_{i}^{n}

a space‐time polynomial of degree M denoted by q h ( x , t ). Such a procedure has first been developed in the Eulerian framework on fixed grids in and subsequently extended to moving meshes in the ALE context . The local space‐time Galerkin technique does not require any neighbor information and is based on an element‐local weak space‐time formulation of the governing PDE () that reads

{falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} \frac{\partial {boldq}_{h}}{\partial t} d boldx d t + {falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} \nabla \cdot boldF ({boldq}_{h}) d boldx d t = {falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} boldS ((), {boldq}_{h}) d boldx d t,

where θ k = θ k ( x , t ) is a space‐time test function.…”

Section: Methodsmentioning

confidence: 99%

“…The generic conserved variable

A_{i}^{*}

of the candidate solution

{boldQ}_{i}^{*}

is acceptable if it holds

\min_{j \in {scriptV}_{i}} ((), A_{j}^{n}, A_{i}^{n}) ⩽ A_{i}^{*} ⩽ \max_{j \in {scriptV}_{i}} ((), A_{j}^{n}, A_{i}^{n}),

where

{scriptV}_{i}

is the Voronoi neighborhood of cell T i that is composed by all elements T j which share at least one vertex of element T i . If condition () is not satisfied, the conserved variable

A_{i}^{*}

is then checked against the so‐called u2 criterion , which determines if this new extremum is smooth or not. Also, ‘NaN’ values could be produced by the candidate solution, and they are marked as ‘problematic’, i.e.…”

Section: Methodsmentioning

confidence: 99%

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An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri

2016

Numerical Methods in Fluids

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Summary In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tn is connected to the new one at tn + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.

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{boldw}_{h}^{n}

for each element

T_{i}^{n}

. On the contrary, the MOOD approach only needs one reconstruction stencil

{scriptS}_{i}

to derive one final reconstruction polynomial

{boldw}_{h}^{n}

, which is used as is, that is, unlimited . The limiting procedure is carried out a posteriori as follows: at the end of each timestep, the discrete candidate numerical solution

{boldQ}_{i}^{*}

{boldQ}_{i}^{*}

is computed again with a lower polynomial degree.…”

Section: Methodsmentioning

confidence: 99%

“…High order of accuracy in time is then achieved using a local space‐time evolution of the reconstruction polynomials

{boldw}_{h}^{n}

, hence obtaining for each control volume

T_{i}^{n}

{falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} \frac{\partial {boldq}_{h}}{\partial t} d boldx d t + {falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} \nabla \cdot boldF ({boldq}_{h}) d boldx d t = {falsefalse}_{\int}^{t^{n}} \int_{T_{i} (t)} θ_{k} boldS ((), {boldq}_{h}) d boldx d t,

where θ k = θ k ( x , t ) is a space‐time test function.…”

Section: Methodsmentioning

confidence: 99%

“…The generic conserved variable

A_{i}^{*}

of the candidate solution

{boldQ}_{i}^{*}

is acceptable if it holds

\min_{j \in {scriptV}_{i}} ((), A_{j}^{n}, A_{i}^{n}) ⩽ A_{i}^{*} ⩽ \max_{j \in {scriptV}_{i}} ((), A_{j}^{n}, A_{i}^{n}),

where

{scriptV}_{i}

is the Voronoi neighborhood of cell T i that is composed by all elements T j which share at least one vertex of element T i . If condition () is not satisfied, the conserved variable

A_{i}^{*}

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

Boscheri

2016

Numerical Methods in Fluids

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“…The Numerical Admissible Detection criteria are based on the Discrete Maximum Principle (DMP) with the so-called relaxed u2 criteria [DLC13,BLD15], a plateau detection and a NaN (Not-A-Number) detection. All these NAD criteria must detect spurious numerical oscillations or lethal situations but ignoring smooth extrema or too small oscillations.…”

Section: Numerical Admissible Detection Criteria (Nad)mentioning

confidence: 99%

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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“…The final timestep Δt is then given by taking Δt = min (Δt C F L , Δt V ). We refer the reader to [13], where we provide all the details needed to carry on the procedure for the timestep evaluation. Figure 10 shows the density distribution as well as the mesh configuration for the Saltzman problem at four different output times.…”

Section: Ale-ader-weno (O3) Qf Ale-ader-weno (O3) Ale-ader-weno (O4) mentioning

confidence: 99%

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

Boscheri

Dumbser

2015

J Sci Comput

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In this paper we present a new and efficient quadrature-free formulation for the family of cell-centered high order accurate direct arbitrary-Lagrangian-Eulerian one-step ADER-WENO finite volume schemes on unstructured triangular and tetrahedral meshes that has been developed by the authors in a recent series of papers (Boscheri et al. in J Comput ). High order of accuracy in time is obtained by using a local space-time Galerkin predictor on moving curved meshes, while a high order accurate nonlinear WENO method is adopted to produce high order essentially non-oscillatory reconstruction polynomials in space. The mesh is moved at each time step according to the solution of a node solver algorithm that assigns a unique velocity vector to each node of the mesh. A rezoning procedure can also be applied when mesh distortions and deformations become too severe. The space-time mesh is then constructed by straight edges connecting the vertex positions at the old time level t n with the new ones at the next time level t n+1 , yielding closed space-time control volumes, on the boundary of which the numerical flux must be integrated. This is done here with a new and efficient quadrature-free approach: the space-time boundaries are split into simplex sub-elements, i.e. either triangles in 2D or tetrahedra in 3D. This leads to space-time normal vectors as well as Jacobian matrices that are constant within each sub-element. Within the space-time Galerkin predictor stage that solves the Cauchy problem inside each element in the small, the discrete solution and the flux tensor are approximated using a nodal space-time basis. Since these space-time basis functions are defined on a reference element and do not change, their integrals over the simplex sub-surfaces of the space-time reference control volume can be integrated once and for all analytically during a preprocessing step. The resulting integrals are then used together B M. Dumbser 123 J Sci Comput with the space-time degrees of freedom of the predictor in order to compute the numerical flux that is needed in the finite volume scheme. We apply the high order algorithm presented in this paper to the equations of hydrodynamics obtaining convergence rates up to fourth order of accuracy in space and time. A set of classical Lagrangian test problems has been solved and the results have been compared with the ones given by the original formulation of the algorithm (Boscheri and Dumbser 2013, 2014). The efficiency has been monitored and measured for each test case and the new quadrature-free schemes were up to 3.7 times faster than the ones based on Gaussian quadrature.Keywords Arbitrary-Lagrangian-Eulerian (ALE) · Finite volume schemes · Quadraturefree flux integration · WENO reconstruction on moving unstructured meshes · High order of accuracy in space and time · Local rezoning · Hydrodynamics

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Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

Cited by 60 publications

References 117 publications

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

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

An Efficient Quadrature-Free Formulation for High Order Arbitrary-Lagrangian–Eulerian ADER-WENO Finite Volume Schemes on Unstructured Meshes

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