High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

Liu, Wei; Cheng, Juan; Shu, Chi‐Wang

doi:10.1016/j.jcp.2009.09.001

Cited by 68 publications

(91 citation statements)

References 30 publications

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“…A strong shock wave propagates from the piston towards the right-hand Table 1 Numerical convergence results for the equations of hydrodynamics using the two-dimensional quadrature-free ALE-ADER-WENO finite volume schemes presented in this article The error norms refer to the variable ρ (density) at time t = 1.0 for first up to fourth order of accuracy The error norms refer to the variable ρ (density) at time t = 1.0 for first up to fourth order of accuracy side of the domain, highly compressing those elements which lie near the piston. This test case is usually proposed in literature [58,62] because it allows the robustness of Lagrangian schemes to be properly verified and we refer the reader to the aforementioned papers for the initialization of this challenging test problem. The exact solution is available and can be easily computed as explained in [9,82].…”

Section: The Saltzman Problemmentioning

confidence: 99%

“…As suggested in [58], the initial Courant number is set to CFL = 0.01 in order to satisfy the geometric CFL condition, hence avoiding invalid elements that may appear due to the very fast motion of the piston into a fluid that is initially at rest. The Courant number is then increased to its usual value of CFL = 0.5 at time t = 0.01.…”

Section: The Saltzman Problemmentioning

confidence: 99%

“…In [18,58] Cheng and Shu used a third order accurate essentially non-oscillatory (ENO) reconstruction operator to develop the first high order Godunov-type Lagrangian finite volume schemes ever. High order accurate one-dimensional Lagrangian ADER finite volume schemes have been presented in [19] and [41].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Saltzman Problemmentioning

confidence: 99%

Section: The Saltzman Problemmentioning

confidence: 99%

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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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“…The first high order Godunov-type Lagrangian finite volume schemes were proposed in [28,81], where a third order accurate essentially non-oscillatory (ENO) reconstruction operator has been employed on moving curved structured meshes.…”

Section: Introductionmentioning

confidence: 99%

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

Boscheri

Loubère

Dumbser

2015

Journal of Computational Physics

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“…More recently, Dumbser developed the ADER discontinuous Galerkin method for solving the linear aeroacoustics systems, where also all the necessary details for the Cauchy-Kovalewski procedure for the compressible Euler equations in multiple space dimensions are given [17]. The Lax-Wendroff type time discretization was also used in high order finite volume schemes, the discontinuous Galerkin method and finite difference schemes [18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

The high order control volume discontinuous Petrov–Galerkin finite element method for the hyperbolic conservation laws based on Lax–Wendroff time discretization

Zhao

2015

Applied Mathematics and Computation

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High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

Cited by 68 publications

References 30 publications

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

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

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

The high order control volume discontinuous Petrov–Galerkin finite element method for the hyperbolic conservation laws based on Lax–Wendroff time discretization

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