Abstract. A method is presented for constructing effective stochastic models for the timeevolution of spatial averages in finite-difference discretizations of partial differential equations. This method relies on the existence of a time-scale separation in the dynamics of the spatial averages and fine-grid variables. The spatial averages, thus, are treated as the slow variables in the system an...

A freely cooling granular gas with a velocity-dependent restitution coefficient is studied in one dimension. The restitution coefficient becomes near elastic when the relative velocity of the colliding particles is less than a velocity scale delta . Different statistical quantities, namely, density distribution, occupied and empty cluster length distributions, and spatial density and velocity c...

Consider an inviscid Burgers equation whose initial data is a Lévy α−stable process Z with α > 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski [13]. Along the way, we contradict a ...

We solve the rotational abrasion model of Roth, Marques, and Durian [Phys. Rev. E 83, 031303 (2011)], a one-dimensional quasilinear partial differential equation resembling the inviscid Burgers equation with the unusual feature of a step function factor as a coefficient. The complexity of the solution is primarily in keeping track of the cases in the piecewise function that results from certain...

This paper investigates different high order finite difference schemes and their accuracy for Burgers equation and Navier-Stokes equations. On a coarse grid, theoretical and numerical analysis indicate that a higher order difference scheme does not necessarily obtain more accurate solutions than a lower order scheme in the regions of high gradient variation (2nd order derivative). On a coarse g...

In this paper we control the first moment of the initial approximations and obtain the order of convergence and the asymptotic profile of a general solution by two explicit “canonical” approximations: a diffusive N-wave and a diffusion wave solution. The order of convergence of both approximations is O(t1/(2r)−3/2) in Lr norm, 1 ≤ r ≤ ∞, as t → ∞, which is faster than the well-known classical c...

In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier–Padé–Galerkin and Fourier–Padé collocation methods by expressin...