We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31, pp. 334–368, 2008] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL conditio...

The logically rectangular finite volume grids for two-dimensional partial differential equations on a sphere and for three-dimensional problems in a spherical shell introduced recently have nearly uniform cell size, avoiding severe Courant number restrictions. We present recent results with adaptive mesh refinement using the GeoClaw software and demonstrate well-balanced methods that exactly ma...

Moist multi-scale models for the hurricane embryo ANDREW J. MAJDA1†, YULONG XING AND MAJID MOHAMMADIAN Department of Mathematics and Climate, Atmosphere and Ocean Science, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Mathematics, University...

Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [8, 17, 18]. These methods work in a satisfactory way either in rarefied or fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper...

There are numerous min-max principles about the eigenvalues of a Hermitian matrix. The most general ones are Wielandt’s min-max principles which include the Courant-Fisher min-max principles and the trace minimization principles as special cases. In this paper, various extensions of Wielandt’s principles are obtained for a positive semi-definite pencil A − λB by which we mean that A and B are H...

This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different time-steps to be used in different parts of the spatial domain. The discretization is second order accurate in time and preserves the conservation and stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global time-steps (restricted by...

In this paper we develop non-linear ADER schemes for time-dependent scalar linear and non-linear conservation laws in one, two and three space dimensions. Numerical results of schemes of up to fifth order of accuracy in both time and space illustrate that the designed order of accuracy is achieved in all space dimensions for a fixed Courant number and essentially non-oscillatory results are obt...