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# A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equation

Institute of Applied Physics and Computational Mathematics, 100088, Beijing, China; Division of Applied Mathematics, Brown University, 02912, Providence, RI, USA

COMMUNICATIONS IN COMPUTATIONAL PHYSICS Commun. Comput. Phys 12/2008; 4:1008-1024. - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents a new high-order cell-centered Lagrangian scheme for two-dimensional compressible flow. The scheme uses a fully Lagrangian form of the gas dynamics equations, which is a weakly hyperbolic system of conservation laws. The system of equations is discretized in the Lagrangian space by discontinuous Galerkin method using a spectral basis. The vertex velocities and the numerical fluxes through the cell interfaces are computed consistently in the Eulerian space by virtue of an improved nodal solver. The nodal solver uses the HLLC approximate Riemann solver to compute the velocities of the vertex. The time marching is implemented by a class of TVD Runge–Kutta type methods. A new HWENO (Hermite WENO) reconstruction algorithm is developed and used as limiters for RKDG methods to maintain compactness of RKDG methods. The scheme is conservative for the mass, momentum and total energy. It can maintain high-order accuracy both in space and time, obey the geometrical conservation law, and achieve at least second order accuracy on quadrilateral meshes. Results of some numerical tests are presented to demonstrate the accuracy and the robustness of the scheme.Journal of Computational Physics 01/2011; · 2.14 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The intent of the present work was the development of a high-order discontinuous Galerkin scheme for solving the gas dynamics equations written under total Lagrangian form on twodimensional unstructured grids. To achieve this goal, a progressive approach has been used to study the inherent numerical di�culties step by step. Thus, discontinuous Galerkin schemes up to the third order of accuracy have �rstly been implemented for the one-dimensional and two-dimensional scalar conservation laws on unstructured grids. The main feature of the presented DG scheme lies on the use of a polynomial Taylor basis. This particular choice allows in the two-dimensional case to take into general unstructured grids account in a uni�ed framework. In this frame, a vertex-based hierarchical limitation which preserves smooth extrema has been implemented. A generic form of numerical uxes ensuring the global stability of our semi-discrete discretization in the L2 norm has also been designed. Then, this DG discretization has been applied to the one-dimensional system of conservation laws such as the acoustic system, the shallow-water one and the gas dynamics equations system written in the Lagrangian form. Noticing that the application of the limiting procedure, developed for scalar equations, to the physical variables leads to spurious oscillations, we have described a limiting procedure based on the characteristic variables. In the case of the one-dimensional gas dynamics case, numerical uxes have been designed so that our semi-discrete DG scheme satis�es a global entropy inequality. Finally, we have applied all the knowledge gathered to the case of the twodimensional gas dynamics equation written under total Lagrangian form. In this framework, the computational grid is �xed, however one has to follow the time evolution of the Jacobian matrix associated to the Lagrange-Euler ow map, namely the gradient deformation tensor. In the present work, the ow map is discretized by means of continuous mapping, using a �- nite element basis. This provides an approximation of the deformation gradient tensor which satis�es the important Piola identity. The discretization of the physical conservation laws for speci�c volume, momentum and total energy relies on a discontinuous Galerkin method. The scheme is built to satisfying exactly the Geometric Conservation Law (GCL). In the case of the third-order scheme, the velocity �eld being quadratic we allow the geometry to curve. To do so, a Bezier representation is employed to de�ne the mesh edges. We illustrate the robustness and the accuracy of the implemented schemes using several relevant test cases and performing rate convergences analysis.11/2012, Degree: Phd, Supervisor: Pierre-Henri Maire; Rémi Abgrall - [Show abstract] [Hide abstract]

**ABSTRACT:**Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for two-dimensional 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 cell-centered high-order 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 node-based solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to third-order 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.Journal of Computational Physics. 01/2014; 276:188–234.

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