Showing papers by "Lorenzo Pareschi published in 2000"

Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

Abstract: Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work [Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws [Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277]. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.

Numerical Solution of the Boltzmann Equation I: Spectrally Accurate Approximation of the Collision Operator

Abstract: In this paper we show that the use of spectral Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with infinite-order accuracy momentum and energy. Consistency of the method is also proved, and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method.

Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations

Abstract: We present new implicit-explicit (IMEX) Runge Kutta methods suitable for time dependent partial differential systems which contain stiff and non stiff terms (i.e. convection-diffusion problems, hyperbolic systems with relaxation). Here we restrict to diagonally implicit schemes and emphasize the relation with splitting schemes and asymptotic preserving schemes. Accuracy and stability properties of these schemes are studied both analytically and numerically.

Numerical Schemes for Hyperbolic Systems of Conservation Laws with Stiff Diffusive Relaxation

Abstract: Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve, and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.

On the stability of spectral methods for the homogeneous Boltzmann equation

Abstract: In this paper we introduce a new smoothed scheme derived from the spectral Fourier method for the homogeneous Boltzmann equation recently introduced in [14, 15]. More precisely we show that using suitable smoothing filters the method can be designed in such a way that the spectral solution remains positive in time and preserves the total mass. Several numerical examples are given to illustrate the previous analysis.

Asymptotic preserving Monte Carlo methods for the Boltzmann equation

Abstract: We consider some recently developed unconditionally stable numerical schemes for the Boltzmann equation, called Time Relaxed (TR) schemes. They share the important property of providing the correct fluid dynamic limit. Stability analysis of the schemes is performed, and the A-stability and L-stability of the schemes is studied. Monte Carlo methods based on TR discretizations are briefly reviewed. In particular, first and second order particle schemes are compared with a hybrid scheme, in which the equilibrium part of the distribution is described analytically.

Méthode spectrale rapide pour l'équation de Fokker–Planck–Landau

Abstract: Resume Dans cette Note, nous introduisons une nouvelle methode spectrale pour l'equation de Fokker–Planck–Landau (FPL) La methode permet, contrairement au cout usuel en O( n 2 ) de l'operateur de collision de FPL, d'obtenir des solutions numeriques spectralement precises avec simplement O( n log 2 n ) operations

Central schemes for hydrodynamical limits of discrete-velocity kinetic models

Abstract: The hydrodynamical scalings of many discrete-velocity kinetic models lead to a small-relaxation time behavior governed by the corresponding Euler type hyperbolic equations or Navier-Stokes type parabolic equations. Using as a prototype a simple discrete-velocity model of the Boltzmann equation we develop a class of central schemes with the correct asymptotic limit that work with uniform second order accuracy with respect to the scaling parameter. Numerical results for both the fluid-dynamic limit and the diffusive limit show the robustness of the present approach.