We propose two-dimensional central finite volume methods based on our multidimensional extensions of Nessyahu and Tadmor's one-dimensional non-oscillatory central scheme and a constrained transport-type method to solve ideal magnetohydrodynamic problems (MHD) and shallow water magnetohydrodynamic problems (SMHD). The main numerical scheme is second-order accurate both in space and time and uses an original Cartesian grid coupled to a Cartesian-or diamond-staggered dual grid to by-pass the resolution of the Riemann problems at the cell interfaces. To treat the non-vanishing magnetic field/flux divergence we have constructed an adaptation of Evans and Hawley's constrained transport method specifically designed for central schemes. Our numerical results show the efficiency and the potential of the scheme. Copyright (C) 2010 John Wiley & Sons, Ltd.

In this paper, we introduce a detector of discontinuities (DoD), based on the entropy production rates over two complementary meshes. This DoD has the capability to differentiate between rarefaction waves, contact discontinuities and shocks, and takes full advantage of leading edge computer technology. Combining it with Harten's artificial compression method (ACM), we get a couple ACM/DoD, independent from the base central scheme used for main computations, that permits sharp capture of discontinuities. We apply this to the homogeneous Euler system of conservation laws and to the ZND detonation model.

We present second-order accurate central finite volume methods adapted here to three-dimensional problems in ideal magnetohydrodynamics. These methods alternate between two staggered grids, thus leading to Riemann solver-free algorithms with relatively favorable computing times.

Detonations can be modeled by a compressible reactive flow involving a single exothermic reaction between two chemical states: the unburnt gas and the burnt gas. The detonation wave can be modeled as a powerful nonreactive shock followed by a deflagration wave, where the gas is burnt. The shock propagates in the unburnt gas, heating it; if the ignition temperature is reached through this purely mechanical shock, then a chemical reaction is triggered. One differentiates detonations from other types of combustion by the great quantity of energy they release, thus making negligible any potential contribution from viscosity, conduction, or heat radiation. Neglecting these local effects makes the reactive Euler equations the most adequate way to describe a detonation process at large scale.

AbstractWe present second‐order accurate central finite volume methods adapted to three‐dimensional ideal magnetohydrodynamics problems. These methods alternate between two staggered grids, thus leading to Riemann solver‐free algorithms with relatively favorable computing times. The div·B = 0 constraint on the magnetic field is enforced with a suitable adaptation of the constrained transport method to our central schemes. Numerical experiments show the feasibility of the proposed methods and our results are in good agreement with existing results in the recent literature. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Since their appearance in 1990, non-oscillatory central schemes (Nessyahu & Tadmor 1990) continuously gained in popularity for multiple reasons. In (Arminjon V in (Arminjon & Touma 2005), we have presented a new constrained transport approach for central schemes (“CTCS”) in the case of diamond dual cells. In contrast with (Ziegler 2004), our new CTCS approach does not require any staggering of the magnetic field. For simplicity, we present here the 2-dimensional version of our new central scheme with Cartesian dual cells and the corresponding CTCS divergence treatment, and compare, for some challenging problems, both 2-dimensional methods (Cartesian dual cells vs. Diamond dual cells).

Two and three-dimensional finite volume extensions of the Lax–Friedrichs (LF) and Nessyahu–Tadmor one-dimensional difference schemes were previously presented and successfully applied to several problems for nonlinear hyperbolic systems, and in particular to typical test cases for both inviscid and viscous compressible flows. These “central” schemes by-pass the resolution, at the cell interfaces, of the Riemann problems, thanks to the use of the staggered LF scheme which serves as the base scheme on which high order finite volume methods can be constructed using van Leer’s MUSCL-type limited reconstruction principle. For this purpose, two dual grids are used at alternate time steps. These methods are extended here to several problems in one- and multi-dimensional ideal compressible magnetohydrodynamics using a modified version of the first author’s central methods with oblique (diamond shaped) dual cells. In two-dimensions the system has eight equations and solving the corresponding Riemann problem is an elaborate and time-consuming process. Central methods lead to significant computing time reductions, and the numerical experiments presented here suggest the accuracy is quite satisfactory. In order to satisfy the physical constraint ∇·B=0, we have constructed a strategy (“CTCS”) inspired from the Constrained Transport method of Evans and Hawley. The validity of our base scheme and our CTCS approach is clearly confirmed by the results.

A new modified version of the Nessyahu–Tadmor (NT) 1-dimensional finite volume central scheme is presented, as well as corresponding new versions for 2D-structured and 3D-unstructured grids inspired from the NT-scheme. The modification avoids the intermediate predictor time step between tn and tn+1. Although this does not really bring about substantial accuracy/computer time improvements in the 1D case, in the 2- and 3-dimensional cases, the modified scheme does lead to important reductions in computer times. 3D comparative simulations for the shock tube problem and for a supersonic inviscid flow through a channel with a 4% circular bump are presented.

A new modified version of the Nessyahu-Tadmor (NT) 1-dimensional finite volume central scheme is presented, as well as corresponding new versions for 2D-structured and 3D-unstructured grids inspired from the NT scheme. The modification avoids the intermediate predictor time step between t n and t n+1. The CPU gain is not very important in the 1D case, but becomes significant in the 2 and 3D cases. 3D comparative simulations for the shock tube problem and for a supersonic inviscid flow through a channel with a 4% circular bump are presented.

We present a 3D finite volume generalization of the 1-dimensional Lax-Friedrichs and Nessyahu-Tadmor schemes for hyperbolic equations on Cartesian grids. The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is by-passed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-step, 3-dimensional non-oscillatory centered scheme in finite volume formulation.Piecewise linear cell interpolants using several van Leer-type limiting techniques to estimate the gradient (van Leer, van Albada, SuperBee, MinMod) lead to a non-oscillatory spatial resolution of order superior to 1. The fact that the expected second-order resolution is not fully attained in 3D is investigated first by considering an alternate dual grid (in 2D), and by using the original van Albada limiter in 3D.Numerical results for a linear advection problem with continuous and discontinuous initial conditions in 2D and 3D show the accuracy and stability of the method. A comparison is made between the 2D Arminjon-Stanescu-Viallon and Jiang-Tadmor formulations and the new one. A new simple projection method is used for the gradients in the new 2D scheme. We also include results for the 3D Euler system (channel with a forward facing step). (C) 2001 Published by Elsevier Science B.V. on behalf of IMACS.

We present a 3D finite volume generalization of the l-dimensional Lax-Friedrichs and Nessyahu-Tadmor schemes for hyperbolic equations on unstructured tetrahedral grids. The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of the Riemann problem at the cell interfaces is by-passed thanks to the use of the staggered Lax-Friedrichs scheme, is extended here to a two-step, three-dimensional non-oscillatory centered scheme in finite volume formulation. Piecewise linear cell interpolants using Venkatakrishnan's limiter combined with diverse techniques to estimate the gradients lead to a non-oscillatory spatial resolution of order 2. The fact that the expected second order resolution is not fully attained in 3D for nonlinear systems might be caused by the absence of grid adaptation in our calculations. Numerical results for a linear advection problem with continuous initial conditions in 3D show the accuracy and stability of the method. We also include results for the 3D Euler system (shock tube problem).

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, for hyperbolic conservation laws and rectangular or unstructured triangular grids, of two-dimensional finite volume extensions of the Lax-Friedrichs and Nessyahu-Tadmor (Nessyahu, R., and Tadmor, E., "Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws," Journal of Computational Physics, Vol. 87, No. 2, 1990, pp. 408-463) central difference schemes, in which the resolution of Riemann problems at cell interfaces is bypassed thanks to the use of the Lax-Friedrichs scheme on two specific staggered grids. MUSCL-type piecewise linear cell interpolants, slope limiters, and a two-step predictor-corrector time discretization lead to an oscillation-free quasi-second-order resolution. For the viscous terms, we use a centered finite element approximation inspired by Rostand-Stoufflet (Rostand, P., and Stoufflet, B., "Finite Volume Galerkin Methods for Viscous Gas Dynamics:" Institut National de Recherche en Informatique ct en Automatique, INRIA Research Rept. 863, Rocquencourt, France, 1988). To improve the quality of the resolution, we use a grid adaptation algorithm proposed by Castro Diaz and Hecht (Castro Diaz, M., and Hecht, F., "Anisotropic Surface Mesh Generation:" Institut National de Recherche en Informatique et en Automatique, INRIA Research Rept. 2672, Rocquencourt, France, Oct. 1995). Numerical experiments on classical test problems (supersonic viscous flow over a Bat plate, around a NACA 0012 airfoil, around an ellipse, and a HERMES-type double ellipse), including comparison with other methods, lead to fairly competitive results with favorable computing times, very sharp capture of shocks and boundary layers, and accurate simulation of the boundary-layer detachment.

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

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.

The non-oscillatory central difference scheme of Nessyahu and Tadmor, in which the resolution of 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.

The Nessyahu-Tadmor scheme is a van Leer type scheme built on the Lax-Friedrichs scheme. It is defined for a one-dimensional hyperbolic problem and the convergence to the entropic solution is proved in the case of a single equation. In this Note an extension to a two-dimensional scalar equation with an unstructured mesh is proposed.