A class of residual distribution schemes and their relation to relaxation systems

Residual distributions (RD) schemes are a class of of high-resolution finite volume methods for unstructured grids. A key feature of these schemes is that they make use of genuinely multidimensional (approximate) Riemann solvers as opposed to the piecemeal 1D Riemann solvers usually employed by finite volume methods. In 1D, LeVeque and Pelanti [J. Comp. Phys. 172, 572 (2001)] showed that many of the standard approximate Riemann solver methods (e.g., the Roe solver, HLL, Lax-Friedrichs) can be obtained from applying an exact Riemann solver to relaxation systems of the type introduced by Jin and Xin [Comm. Pure Appl. Math. 48, 235 (1995)]. In this work we extend LeVeque and Pelanti's results and obtain a multidimensional relaxation system from which multidimensional approximate Riemann solvers can be obtained. In particular, we show that with one choice of parameters the relaxation system yields the standard N-scheme. With another choice, the relaxation system yields a new Riemann solver, which can be viewed as a genuinely multidimensional extension of the local Lax-Friedrichs scheme. This new Riemann solver does not require the use Roe-Struijs-Deconinck averages, nor does it require the inversion of an m-by-m matrix in each computational grid cell, where $m$ is the number of conserved variables. Once this new scheme is established, we apply it on a few standard cases for the 2D compressible Euler equations of gas dynamics. We show that through the use of linear-preserving limiters, the new approach produces numerical solutions that are comparable in accuracy to the N-scheme, despite being computationally less expensive.Comment: 46 pages, 14 figure

Solving the Boltzmann equation in N log N

In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension three. These algorithms are implemented for the solution of the Boltzmann equation in two and three dimension, first for homogeneous solutions, then for general non homogeneous solutions. The results are compared to explicit solutions, when available, and to Monte-Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte-Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].Comment: 32 page

Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions

We demonstrate how meshfree finite difference methods can be applied to solve vector Poisson problems with electric boundary conditions. In these, the tangential velocity and the incompressibility of the vector field are prescribed at the boundary. Even on irregular domains with only convex corners, canonical nodal-based finite elements may converge to the wrong solution due to a version of the Babuska paradox. In turn, straightforward meshfree finite differences converge to the true solution, and even high-order accuracy can be achieved in a simple fashion. The methodology is then extended to a specific pressure Poisson equation reformulation of the Navier-Stokes equations that possesses the same type of boundary conditions. The resulting numerical approach is second order accurate and allows for a simple switching between an explicit and implicit treatment of the viscosity terms.Comment: 19 pages, 7 figure

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp. energy estimates of Section 7, corrected bad forward references, expanded Remark 1.17, end of introductio