A cell-centered Lagrangian scheme with the preservation of symmetry and conservation properties for compressible fluid flows in two-dimensional cylindrical geometry

SUMMARYIn this paper we present a class of high order accurate cell-centered Arbitrary-Eulerian-Lagrangian (ALE) one-step ADER-WENO finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes along the principles laid out in [1]. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space-time Galerkin predictor allows the schemes to be high order accurate also in time by using an element-local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained i) either as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu [2], or, ii) as a solution of multiple one-dimensional half-Riemann problems around a vertex, as suggested by Maire [3], or, iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional HLL Riemann solver recently proposed by Balsara et al. [4]. Once the vertex velocity and thus the new node location has been determined by the node solver, the local mesh motion 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 . If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space-time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unneccesary, since the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic MHD equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration.

scriptN {scriptS}_{cs}

using the idea of Cheng and Shu, as was carried out in , but taking a mass weighted average velocity among the neighborhood

{scriptV}_{k}

of node k , that is,

{\overset{boldV}{false}}_{k}^{n} = \frac{1}{μ_{k}} \sum_{T_{j}^{n} \in {scriptV}_{k}} μ_{k, j} {\overset{boldV}{false}}_{k, j},

with

μ_{k} = \sum_{T_{j}^{n} \in {scriptV}_{k}} μ_{k, j}, μ_{k, j} = ρ_{j}^{n} | T_{j}^{n} | .

The local weights μ k , j , which are the masses of the elements

T_{j}^{n}

, are defined by multiplying the cell averaged value of density

ρ_{j}^{n}

with the cell area

| T_{j}^{n} |

, while the local velocity contributions

{\overset{boldV}{false}}_{k, j}

are computed by integrating in time the high‐order vertex‐extrapolated velocity at node k as

{\overset{boldV}{false}}_{k, j} = ((), {falsefalse}_{\int}^{0})

…”

Section: Numerical Schemementioning

confidence: 99%

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

Boscheri

Dumbser

Balsara

2014

“…The mesh motion, that is, the computation of a unique velocity vector

{boldV}_{k}^{n}

for each vertex k , is carried out using the simple and general nodal solver of Cheng and Shu with the modifications introduced in . The final node velocity is given as a mass weighted average among all the contributions V k , j coming from the neighbor elements T j , that is,

{boldV}_{k}^{n} = \frac{1}{μ_{k}} \sum_{T_{j}^{n} \in {scriptV}_{k}} μ_{k, j} {boldV}_{k, j},

with the local weights defined as the product between the cell averaged value of density

ρ_{j}^{n}

and the cell volume

| T_{j}^{n} |

μ_{k} = \sum_{T_{j}^{n} \in {scriptV}_{k}} μ_{k, j}, μ_{k, j} = ρ_{j}^{n} | T_{j}^{n} | .

For each vertex k , we consider its Voronoi neighborhood

{scriptV}_{k}

, which is composed by all the neighbors that share the common node k , and for each neighbor we evaluate its local contribution V k , j by integrating in time the high‐order vertex‐extrapolated velocity at node k as

{boldV}_{k, j} = ((), {falsefalse}_{\int}^{0} θ_{l} (ξ_{m (k)}^{e}, η_{m (k)}^{e}, ζ_{m (k)}^{e}, τ) d τ) fals...

…”

Section: Methodsmentioning

confidence: 99%

“…The mesh motion, that is, the computation of a unique velocity vector V n k for each vertex k, is carried out using the simple and general nodal solver of Cheng and Shu [41,43,44] with the modifications introduced in [48]. The final node velocity is given as a mass weighted average among all the contributions V k;j coming from the neighbor elements T j , that is,…”

Section: Mesh Motionmentioning

confidence: 99%

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

Boscheri

2016

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.

“…Equivalence conditions are used to devise a class of FV schemes, in which all grid-dependent quantities are defined in terms of FE integrals. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].The present approach moves from the mixed FV/finite element (FE) approach [16], which has been already successfully applied to the multidimensional Cartesian case [17,18] and the cylindrical case with axial symmetry [19], within the throughflow approximation [20] and more recently, in a unified fashion form for orthogonal coordinate systems [21]. The two-dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived.…”

mentioning

confidence: 99%

“…More recently, Maire proposed a cell-centred Lagrangian formulation to capture axisymmetrical, that is, spherical, imploding shock waves [11]; Illenseer and Duschl generalized the two-dimensional central upwind to curvilinear grids [12]; Clain and collaborators extended a multi-slope MUSCL scheme to cylindrical coordinates [13]. In the framework of Lagrangian or ALE solvers, the explosion and implosion problems in two spatial dimensions were solved by Cheng and Shu using a cell-centred Lagrangian scheme [14]; Loubère and others used the reconnection algorithm for ALE framework in cylindrical coordinates [15].…”

mentioning

confidence: 99%

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

Santis

Geraci

Guardone

2013

SUMMARYA complete set of equivalence conditions, relating the mass‐lumped Bubnov–Galerkin finite element (FE) scheme for Lagrangian linear elements to node‐centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three‐dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid‐dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three‐dimensional cylindrical coordinate system. The two‐dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one‐dimensional simulations and simplified models. Copyright © 2013 John Wiley & Sons, Ltd.