Article
A Third Order Conservative Lagrangian Type Scheme on Curvilinear Meshes for the Compressible Euler Equations
Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA
Communications in Computational Physics (Impact Factor: 1.94). 12/2008; 4(5):10081024. ABSTRACT
Based on the high order essentially nonoscillatory (ENO) Lagrangian type scheme on quadrilateral meshes presented in our earlier work [3], in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics. The main purpose of this work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straightline edges, which restricts the accuracy of the resulting scheme to at most second order. The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes. Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and nonoscillatory properties.
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 "A leastsquares (LSQ) solution for those coefficients involves the computationally expensive solution of a multiequation system for each cell and for each reconstructed variable. Further, as discussed by Cheng and Shu [27], third and higher order schemes require curvature of the cell faces to achieve the specified order. Our work focuses on improvements to the secondorder method while retaining straight cell faces. "
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ABSTRACT: This work presents an extension of a second order cellcentered hydrodynamics scheme on unstructured polyhedral cells toward higher order. The goal is to reduce dissipation, especially for smooth flows. This is accomplished by multiple piecewise linear reconstructions of conserved quantities within the cell. The reconstruction is based upon gradients that are calculated at the nodes, a procedure that avoids the leastsquare solution of a large equation set for polynomial coefficients. Conservation and monotonicity are guaranteed by adjusting the gradients within each cell corner. Results are presented for a wide variety of test problems involving smooth and shockdominated flows, fluids and solids, 2D and 3D configurations, as well as Lagrange, Eulerian, and ALE methods. 
 "To reach a higher order of accuracy, one has to take into account a higher order discretization of the kinematics of the flow. This point has been successfully addressed in [13] in which the authors present a thirdorder Lagrangian scheme for solving gas dynamics equations on curvilinear meshes. The physical variables are computed through the use of a highorder ENO conservative reconstruction, and the determination of the vertex velocity is obtained by means of the conserved variables. "
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ABSTRACT: Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for twodimensional fluid flows on general unstructured grids. Contrary to the updated Lagrangian formulation, which refers to the current moving configuration of the flow, the total Lagrangian formulation refers to the fixed reference configuration, which is usually the initial one. In this framework, the Lagrangian and Eulerian descriptions of the kinematical and the physical variables are related by means of the Piola transformation. Here, we describe a cellcentered highorder DG discretization of the physical conservation laws. The geometrical conservation law, which governs the time evolution of the deformation gradient, is solved by means of a finite element discretization. This approach allows to satisfy exactly the Piola compatibility condition. Regarding the DG approach, it relies on the use of a polynomial space approximation which is spanned by a Taylor basis. The main advantage in using this type of basis relies on its adaptability regardless the shape of the cell. The numerical fluxes at the cell interfaces are computed employing a nodebased solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to thirdorder of our DG method. First, we show its ability to accurately capture geometrical features of a flow region employing curvilinear grids. Second, we demonstrate the dramatic improvement in symmetry preservation for radial flows. 
 "Further, the kinematic velocity field also admits a linear continuous representation. Therefore, as noticed in [11], this approximation of the grid motion implies a secondorder error in the numerical method. To reach a higher order of accuracy, one has to take into account a higher order discretization of the kinematics of the flow. "
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ABSTRACT: Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for twodimensional fluid flows on general unstructured grids. Contrary to the updated Lagrangian formulation, which refers to the current moving configuration of the flow, the total Lagrangian formulation refers to the reference fixed configuration, which is usually the initial one. In this framework, the Lagrangian and Eulerian descriptions of the kinematical and the physical variables are related by means of the Piola transformation. Here, we describe a cellcentered highorder DG discretization of the physical conservation laws. The geometrical conservation law, which governs the time evolution of the deformation gradient, is solved by means of a finite element discretization. This approach allows to satisfy exactly the Piola compatibility condition. Regarding the DG approach, it relies on the use of a polynomial space approximation which is spanned by a Taylor basis. The main advantage in using this type of basis relies on its adaptability regardless the shape of the cell. The numerical fluxes at the cell interfaces are computed employing a nodebased solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to thirdorder of our DG method. First, we show its ability to accurately capture geometrical features of a flow region employing curvilinear grids. Second, we demonstrate the dramatic improvement in symmetry preservation for radial flows.