Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities

International audienceIn this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the Discrete Duality Finite Volume framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems

A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure

In this paper we formulate and test numerically a fully-coupled discontinuous Galerkin (DG) method for incompressible two-phase flow with discontinuous capillary pressure. The spatial discretization uses the symmetric interior penalty DG formulation with weighted averages and is based on a wetting-phase potential / capillary potential formulation of the two-phase flow system. After discretizing in time with diagonally implicit Runge-Kutta schemes the resulting systems of nonlinear algebraic equations are solved with Newton's method and the arising systems of linear equations are solved efficiently and in parallel with an algebraic multigrid method. The new scheme is investigated for various test problems from the literature and is also compared to a cell-centered finite volume scheme in terms of accuracy and time to solution. We find that the method is accurate, robust and efficient. In particular no post-processing of the DG velocity field is necessary in contrast to results reported by several authors for decoupled schemes. Moreover, the solver scales well in parallel and three-dimensional problems with up to nearly 100 million degrees of freedom per time step have been computed on 1000 processors

High-order conservative finite difference GLM-MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. (J. Comput. Phys. 175 (2002) 645-673). The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics (Aug 7 2009

"Mariage des Maillages": A new numerical approach for 3D relativistic core collapse simulations

We present a new 3D general relativistic hydrodynamics code for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. It uses spectral methods for solving the metric equations, assuming the conformal flatness approximation for the three-metric. The matter equations are solved by high-resolution shock-capturing schemes. We demonstrate that the combination of a finite difference grid and a spectral grid can be successfully accomplished. This "Mariage des Maillages" (French for grid wedding) approach results in high accuracy of the metric solver and allows for fully 3D applications using computationally affordable resources, and ensures long term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations. A variety of tests in 2D and 3D is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, demonstrating the ability to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess gravitational waveform extraction, which is based on the quadrupole formula.Comment: 29 pages, 16 figures; added more information about convergence tests and grid setu

Finite volume method for nonlinear transmission problems

International audienceIn this paper we study the approximation of solutions to nonlinear transmission problems in the Discrete Duality Finite Volume framework. We extend the analysis of the m-DDFV schemes introduced in a previous work to more general nonlinear elliptic problems like bimaterial problems arising in elastic-plastic mechanics studied by Liu in 1999