Related papers: Accurate Spectral Numerical Schemes for Kinetic Eq…
In this article, we are interested in the asymptotic analysis of a finite volume scheme for one dimensional linear kinetic equations, with either Fokker-Planck or linearized BGK collision operator. Thanks to appropriate uniform estimates,…
In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments.…
The central object in wave turbulence theory is the wave kinetic equation (WKE), which is an evolution equation for wave action density and acts as the wave analog of the Boltzmann kinetic equations for particle interactions. Despite recent…
We address the problem of approximating the moments of the solution, $\boldsymbol{X}(t)$, of an It\^o stochastic differential equation (SDE) with drift and a diffusion terms over a time-grid $t_0, t_1, \ldots, t_n$. In particular, we assume…
We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…
In this article, families of non-linear subdivision schemes are presented that are based on univariate polynomials up to degree three. Theses families of schemes are constructed by using dynamic iterative re-weighed least squares method.…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
Several relaxation approximations to partial differential equations have been recently proposed. Examples include conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems. The present paper focuses…
Kinetic energy equipartition is a premise for many deterministic and stochastic molecular dynamics methods that aim at sampling a canonical ensemble. While this is expected for real systems, discretization errors introduced by the numerical…
We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension.…
Employing physically-consistent numerical methods is an important step towards attaining robust and accurate numerical simulations. When addressing compressible flows, in addition to preserving kinetic energy at a discrete level, as done in…
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
We propose an approximation of nonlinear renewal equations by means of ordinary differential equations. We consider the integrated state, which is absolutely continuous and satisfies a delay differential equation. By applying the…
We give a new fast method for evaluating sprectral approximations of nonlinear polynomial functionals. We prove that the new algorithm is convergent if the functions considered are smooth enough, under a general assumption on the spectral…
Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto…
We study nonlinear hyperbolic conservation laws with non-convex flux in one space dimension and, for a broad class of numerical methods based on summation by parts operators, we compute numerically the kinetic functions associated with each…
We describe exact kink soliton solutions to nonlinear partial differential equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx} = A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality allows…
In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the…
We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi…
A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided-difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this…