Paul Arminjon

A Finite Volume Extension of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Conservation Laws on Unstructured Grids

The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution or Riemann problems at the cell interfaces is by-passed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-step, two-dimensional non-oscillatory centered scheme in finite volume formulation. The construction of the scheme rests on a finite volume extension of the Lax-Friedrichs scheme, in which the finite volume cells are the barycentric cells constructed around the nodes of an FEM triangulation, for odd time steps, and some quadrilateral cells associated with this triangulation, for even time steps. Piecewise linear cell interpolants using least-squares gradients combined with a van Leer-type slope limiting allow for an oscillation-free second-order resolution. Some preliminary numerical experiments suggest that two-dimensional problems can be handled very efficiently by the method presented here.

Convergence of a Finite Volume Extension of the Nessyahu--Tadmor Scheme on Unstructured Grids for a Two-Dimensional Linear Hyperbolic Equation

The nonoscillatory central difference scheme of Nessyahu and Tadmor is a Godunov-type scheme for one-dimensional hyperbolic conservation laws in which the resolution of Riemann problems at the cell interfaces is bypassed thanks to the use of the staggered Lax--Friedrichs scheme. Piecewise linear MUSCL-type (monotonic upstream-centered scheme for conservation laws) cell interpolants and slope limiters lead to an oscillation-free second-order resolution. Convergence to the entropic solution was proved in the scalar case. After extending the scheme to a two-step finite volume method for two-dimensional hyperbolic conservation laws on unstructured grids, we present here a proof of convergence to a weak solution in the case of the linear scalar hyperbolic equation

Non-Oscillatory Schemes for Multidimensional Euler Calculations with Unstructured Grids

u_t + \divv(\vec V\,u) = 0

A Mixed Finite Volume/Finite Element Method for 2-dimensional Compressible Navier-Stokes Equations on Unstructured Grids

. Since the scheme is Riemann solver--free, it provides a truly multidimensional approach to the numerical approximation of compressible flows, with a firm mathematical basis. Numerical experiments show the feasibility and high accuracy of the method.

Upwind finite volume schemes with anti-diffusion for the numerical study of electric discharges in gas-filled cavities

The purpose of this paper is to present a synthesis of our recent studies related to the design of multi-dimensional non-oscillatory schemes, applying to non-structured finite-element simplicial meshes (triangles, tetrahedra). While the direct utilization of 1-D concepts may produce robust and accurate schemes when applied to non-distorted structured meshes, it cannot when non-structured triangulations are to be used. The subject of the paper is to study the adaptation of the so-called TVD methods to that context. TVD methods have been derived for the design of hybrid first-order/second-order accurate schemes which present in simplified cases monotonicity properties (see, for example, the review [2]). A various collection of first-order accurate schemes can be used, they are derived from an artificial viscosity model or from an approximate Riemann solver. However, the main feature in the design is the choice of the second-order accurate scheme; this choice can rely either on central differencing or on upwind differencing.

Discontinuous finite elements and godunov-type methods for the eulerian equations of gas dynamics

To solve flow problems associated with the Navier-Stokes equations, we construct a mixed finite volume/finite element method for the spatial approximation of the convective and diffusive parts of the flux, respectively. The finite volume component of the method is adapted from the authors’ construction ([1], [2], [3]), for hyperbolic conservation laws and unstructured triangular or rectangular grids, of 2-dimensional finite volume extensions of the Lax-Friedrichs and Nessyahu-Tadmor central difference schemes, in which the resolution of Riemann problems at cell interfaces is by-passed thanks to the use of the Lax-Friedrichs scheme on two specific staggered grids. Piecewise linear cell interpolants, slope limiters and a 2-step time discretization lead to an oscillation-free second order resolution.

Sur un problème d'existence de Lions pour une équation différentielle opérationnelle

Abstract A finite volume method with upwinding and limited anti-diffusion is constructed for the numerical study of electric discharges in gas-filled cavities. We use a physical model presented by Novak and Bartnikas and derived from a similar model introduced by Phelps. The ionization term, as well as the convective and diffusive parts of the particle conservation equations, are treated separately, using a fractional step decomposition. The upwinding used for the convective step is corrected by adjunction of an anti-diffusive term controlled by a Zalesak-type limiter. This guarantees positivity of the particle densities. Numerical experiments performed with this method are in good agreement with previous results, and show that the method allows us to reach much higher densities,for a more accurate simulation of the electric discharge phenomenon.

Construction of TVD-like Artificial Viscosities on Two-Dimensional Arbitrary FEM Grids

Abstract We investigate the numerical solution of typical problems in gas dynamics by discontinuous finite element methods, and compare the results with computations performed with variants of Godunovs conservative method. A one-dimensional shock wave problem with reflection, and a three-dimensional shock tube-type problem with convergent-divergent nozzle geometry are analyzed. For the one-dimensional problem we also present results obtained with a variant of Glimms method. In one dimension, finite elements give valuable results, although they need a substantially larger computing time; in three space dimensions discontinuous elements appear to be too cumbersome, in the present form, to lend themselves to an efficient treatment of time-dependent shock wave problems.

A finite element method for Burgers' equation in hydrodynamics

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