Related papers: Accurate Spectral Numerical Schemes for Kinetic Eq…
The advection-diffusion and wave equations are the fundamental equations governing any physical law and therefore arise in many areas of physics and astrophysics. For complex problems and geometries, only numerical simulations can give…
Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization…
We propose a discrete lattice version of the Fokker-Planck kinetic equation along lines similar to the Lattice-Boltzmann scheme. Our work extends an earlier one-dimensional formulation to arbitrary spatial dimension $D$. A generalized…
Discrete gradient methods are a powerful tool for the time discretization of dynamical systems, since they are structure-preserving regardless of the form of the total energy. In this work, we discuss the application of discrete gradient…
In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization.…
Explicit numerical finite difference schemes for partial differential equations are well known to be easy to implement but they are particularly problematic for solving equations whose solutions admit shocks, blowups and discontinuities.…
Wave Kinetic Equations (WKEs) are often used to describe the evolution of ensemble averaged wave amplitudes for nonlinear wave systems. In the present manuscript we describe a new approach to direct numerical simulation of solutions to…
Moment estimation for stochastic differential equations (SDEs) is fundamental to the formal reasoning and verification of stochastic dynamical systems, yet remains challenging and is rarely available in closed form. In this paper, we study…
In [8], some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, kinetic equations, and Schr{\"o}dinger type equations with a rotation term. In this work, these exact…
The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a nonuniform polynomial dichotomy and that the…
The numerical analysis for the small amplitude motion of an elastic beam with internal damping is investigated in domain with moving ends. An efficient numerical method is constructed to solve this moving boundary problem. The stability and…
This paper presents a novel approach for numerical solution of a class of fourth order time fractional partial differential equations (PDE's). The finite difference formulation has been used for temporal discretization, whereas, the space…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of…
We develop an efficient numerical scheme for the 3D mean-field spherical dynamo equation. The scheme is based on a semi-implicit discretization in time and a spectral method in space based on the divergence-free spherical harmonic…
A discretization scheme is introduced for a set of convection-diffusion equations with a non-linear reaction term, where the convection velocity is constant for each reactant. This constancy allows a transformation to new spatial variables,…
The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…
Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows…
This work presents a novel family of well-balanced numerical schemes for hyperbolic systems of balance laws based on the kinetic relaxation approach. The method begins by transforming the original non-linear system into a linearized kinetic…
We analytically solve for the time dependent solutions of various density evolution models. With specific forms of the diffusion, drift and sink coefficients, the eigenfunctions can be expressed in terms of hypergeometric functions. We…