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The equations of Lagrangian gas dynamics fall into the larger class of overdetermined hyperbolic and thermodynamically compatible (HTC) systems of partial differential equations. They satisfy an entropy inequality (second principle of thermodynamics) and conserve total energy (first principle of thermodynamics). The aim of this work is to construct a novel thermodynamically compatible cell-centered Lagrangian finite volume scheme on unstructured meshes. Unlike in existing schemes, we choose to directly discretize the entropy inequality, hence obtaining total energy conservation as a consequence of the new thermodynamically compatible discretization of the other equations. First, the governing equations are written in fluctuation form. Next, the noncompatible centered numerical fluxes are corrected according to the approach recently introduced by Abgrall et al. using a scalar correction factor that is defined at the nodes of the grid. This perfectly fits into the formalism of nodal solvers which is typically adopted in cell-centered Lagrangian finite volume methods. Semidiscrete entropy conservative and entropy stable Lagrangian schemes are devised, and they are adequately blended together via a convex combination based on either a priori or a posteriori detectors of discontinuous solutions. The nonlinear stability in the energy norm is rigorously demonstrated, and the new schemes are provably positivity preserving for density and pressure. Furthermore, they exhibit zero numerical diffusion for isentropic flows while still being nonlinearly stable. The new schemes are tested against classical benchmarks for Lagrangian hydrodynamics, assessing their convergence and robustness and comparing their numerical dissipation with classical Lagrangian finite volume methods. [ABSTRACT FROM AUTHOR]

Currently, nearly all positivity preserving discontinuous Galerkin (DG) discretizations of partial differential equations are coupled with explicit time integration methods. Unfortunately, for many problems this can result in severe time-step restrictions. The techniques used to develop explicit positivity preserving DG discretizations cannot, however, easily be combined with implicit time integration methods. In this paper, we therefore present a new approach. Using Lagrange multipliers, the conditions imposed by the positivity preserving limiters are directly coupled to a DG discretization combined with a diagonally implicit Runge--Kutta time integration method. The positivity preserving DG discretization is then reformulated as a Karush--Kuhn--Tucker (KKT) problem, which is frequently encountered in constrained optimization. Since the limiter is only active in areas where positivity must be enforced, it does not affect the higher order DG discretization elsewhere. The resulting nonsmooth nonlinear algebraic equations have, however, a different structure compared to most constrained optimization problems. We therefore develop an efficient active set semismooth Newton method that is suitable for the KKT formulation of time-implicit positivity preserving DG discretizations. Convergence of this semismooth Newton method is proven using a specially designed quasi-directional derivative of the time-implicit positivity preserving DG discretization. The time-implicit positivity preserving DG discretization is demonstrated for several nonlinear scalar conservation laws, which include the advection, Burgers, Allen--Cahn, Barenblatt, and Buckley--Leverett equations. [ABSTRACT FROM AUTHOR]

This paper investigates the approximation of the shallow water equations with topography and friction, using continuous finite elements. A new, second-order, parameter-free, well-balanced and positivity preserving explicit approximation technique is introduced. The novelties of the method are the explicit treatment of the friction term, the robust approximation of dry states, a commutator-based, high-order, entropy viscosity, and a local limiting procedure. The computational method is illustrated on various benchmark tests. [ABSTRACT FROM AUTHOR]

Maximum principle or positivity-preserving property holds for many mathematical models. When the models are approximated numerically, it is preferred that these important properties can be preserved by numerical discretizations for the robustness and the physical relevance of the approximate solutions. In this paper, we investigate such d iscretizations of high order accuracy within the central discontinuous Galerkin framework. More specifically, we design and analyze high order maximum-principle-satisfying central discontinuous Galerkin methods for scalar conservation laws, and high order positivity-preserving central discontinuous Galerkin for compressible Euler systems. The performance of the proposed methods will be demo nstrated through a set of numerical experiments. [ABSTRACT FROM AUTHOR]

The VFRoe scheme has been recently introduced by Buffard, Gallouët, and Hérard [Comput. Fluids, 29 (2000), pp. 813-847] to approximate the solutions of the shallow water equations. One of the main interests of this method is to be easily implemented. As a consequence, such a scheme appears as an interesting alternative to other more sophisticated schemes. The VFRoe methods perform approximate solutions in good agreement with the expected ones. However, the robustness of this numerical procedure has not been proposed. Following the ideas introduced by Jin and Xin [Comm. Pure Appl. Math., 45 (1995), pp. 235-276], a relevant relaxation method is derived. The interest of this relaxation scheme is twofold. In the first hand, the relaxation scheme is shown to coincide with the considered VFRoe scheme. In the second hand, the robustness of the relaxation scheme is established, and thus the nonnegativity of the water height obtained involving the VFRoe approach is ensured. Following the same idea, a family of relaxation schemes is exhibited. Next, robust high order slope limiter methods, known as MUSCL reconstructions, are proposed. The final scheme is obtained when considering the hydrostatic reconstruction to approximate the topography source terms. Numerical experiments are performed to attest the interest of the procedure. [ABSTRACT FROM AUTHOR]

The performance of the MUSCL­Hancock upwind scheme is examined on problems of two-dimensional advection. It is found that when the advection direction is skewed relative to the mesh, most discrete total variation (TVD) limiters give rise to large spurious oscillations near discontinuities. The cause of these oscillations is traced to reconstructions that are not bounded by neighboring cell averages. It is proved that if the reconstruction in each cell is bounded by the cell averages of first order neighbors, then the MUSCL­Hancock scheme is positivity preserving. A simple limiter that achieves such bounded reconstructions is presented next along with a variant that is uniformly second order accurate. Numerical experiments show that the new schemes are accurate and efficient and compare favorably with other schemes. [ABSTRACT FROM AUTHOR]