A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

Pierre-Henri Maire
2009 Journal of Computational Physics  
We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total
more » ... ergy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme. discretize not only the gas dynamics equations but also the vertex motion in order to move the mesh. Moreover, the numerical fluxes of the physical conservation laws must be determined in a compatible way with the vertex velocity so that the geometric conservation law (GCL) is satisfied, namely the rate of change of a Lagrangian volume has to be computed coherently with the node motion. This critical requirement is the cornerstone of any Lagrangian multi-dimensional scheme. The most natural way to solve this problem employs a staggered discretization in which position, velocity and kinetic energy are centered at points, while density, pressure and internal energy are within cells. The dissipation of kinetic energy into internal energy through shock waves is ensured by an artificial viscosity term. Since the seminal works of von Neumann and Richtmyer [42] , and Wilkins [43], many developments have been made in order to improve the accuracy and the robustness of staggered hydrodynamics [11, 9, 7] . More specifically, the construction of a compatible staggered discretization leads to a scheme that conserves total energy in a rigorous manner [10, 8] . We note also the recent development of a variational multi-scale stabilized approach in finite element computation of Lagrangian hydrodynamics, where a piecewise linear approximation was adopted for the variables [35, 34] . The case of Q1/P0 finite element is studied in [36] , where the kinematic variables are represented using a piecewise linear continuous approximation, while the thermodynamic variables utilize a piecewise constant representation. An alternative to the previous discretizations is to derive a Lagrangian scheme based on the Godunov method [20] . In comparison to staggered discretizations, Godunov-type methods exhibit the good property of being naturally conservative, they do not need an artificial viscosity and they allow a straightforward implementation of conservative remapping methods when they are used in the context of the Arbitrary Lagrangian Eulerian (ALE) strategy. In the Godunov-type method approach, all conserved quantities, including momentum, and hence cell velocity are cell-centered. The cell-face quantities, including a face-normal component of the velocity, are available from the solution of an approximate Riemann problem at each cell face. However, it remains to determine the vertex velocity in order to move the mesh. In [1], Dukowicz has proposed to use a weighted least squares algorithm to compute the vertex velocity by requiring that the vertex velocity projected in the direction of a face normal should equal the Riemann velocity on that face. It turns out that this algorithm is capable of generating additional spurious components in the vertex velocity field. Hence, it leads to an artificial grid motion which requires a very expensive treatment [18] . This default comes probably from the fact that the flux computation is not compatible with the node displacement, and hence the GCL is not satisfied. An important achievement concerning the compatibility between flux discretization and vertex velocity computation has been introduced by Després and Mazeran [16] . In this paper, they present a scheme in which the interface fluxes and the node velocity are computed coherently thanks to an approximate Riemann solver located at the nodes. This original approach leads to a first-order conservative scheme which satisfies a local semidiscrete entropy inequality. The multi-dimensional high-order extension of this scheme is developed in [12] . A thorough study of the properties of the Després-Mazeran nodal solver shows a strong sensitivity to the cell aspect ratio, refer to [28] , which can lead to severe numerical instabilities. This drawback is critical for real-life Lagrangian computations in which the grid often contains high aspect ratio cells. To overcome this difficulty, Maire et al. [28] have proposed an alternative scheme that successfully solves the aspect ratio problem and keeps the compatibility between fluxes discretization and vertices velocity computation. This first-order scheme also conserves momentum, total energy, and fulfills a local entropy inequality. Its main feature lies in the discretization of the pressure gradient, which is designed using two pressures at each node of a cell, each nodal pressure
doi:10.1016/j.jcp.2008.12.007 fatcat:h6xevxf5svgfnjzf6ed75xoyte