We develop a simple energy method for proving the stability of finite difference schemes for multidimensional hyperbolic initial-boundary value problems. In particular, we extend to several space dimensions and to variable coefficients a crucial stability result by Goldberg and Tadmor for Dirichlet boundary conditions. This allows us to give some conditions on the discretized operator that ensu...

In this paper, we consider several high order schemes in one space dimension. In particular, we compare the second order relaxation (<<1) or "relaxed" (=0) schemes of Jin-Xin 4], with the second order Lax-Friedrichs scheme of Nessyahu-Tadmor 6], and with higher order ENO and WENO schemes. This comparison is rst made on Sod shock tube, and then on a very pathological example of a p-system constr...

We show the discrete lip+-stability for a relaxation scheme proposed by Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 235–277] to approximate convex conservation laws. Equipped with the lip+-stability we obtain global error estimates in the spaces W s,p for −1 ≤ s ≤ 1/p, 1 ≤ p ≤ ∞ and pointwise error estimates for the approximate solution obtained by the relaxation scheme. The proof uses ...

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers’ equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two...

Central schemes may serve as universal finite-difference methods for solving nonlinear convection–diffusion equations in the sense that they are not tied to the specific eigenstructure of the problem, and hence can be implemented in a straightforward manner as black-box solvers for general conservation laws and related equations governing the spontaneous evolution of large gradient phenomena. T...

Title of dissertation: MULTISCALE PROBLEMS ON COLLECTIVE DYNAMICS AND IMAGE PROCESSING: THEORY, ANALYSIS AND NUMERICS Changhui Tan, Doctor of Philosophy, 2014 Dissertation directed by: Professor Eitan Tadmor Department of Mathematics Institute for Physical Science & Technology Center for Scientific Computation and Mathematical Modeling Multi-scale problems appear in many contexts. In this thesi...

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton–Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241–282; A. Kurganov and D. Levy, SIAM J. Sc...

We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn Serrin (1985), which can be used as a phase transition model. Compared the classical Navier-Stokes equations, there is smoothing effect on density that comes from terms. First, we prove global solutions critical regularity have been constructed in [11] second author B. Desjardins (2001),...

In this work, we discuss kinetic descriptions of flocking models, of the so-called CuckerSmale [4] and Motsch-Tadmor [10] types. These models are given by Vlasov-type equations where the interactions taken into account are only given long-range bi-particles interaction potential. We introduce a new exact rescaling velocity method, inspired by the recent work [6], allowing to observe numerically...