arXiv subjects

Murtazo Nazarov

Publications and source records attributed to Murtazo Nazarov.

21 records · Page 2Linked to original sources

arXiv2020

Regularity of almost periodic solutions of Poisson's equation

This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-\Delta u = f$ in $\mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poisson's equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poisson's equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $\partial u/ \partial x_i$, $i=1, \ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Green's function for Poisson's equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.

math.AP↗

arXiv2020

Numerical Simulations of Surface-Quasi Geostrophic Flows on Periodic Domains

We propose a novel algorithm for the approximation of surface-quasi geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed and the algorithm does not have restrictions on the mesh structure nor on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second-order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.

math.NA↗

arXiv2017

Second-order invariant domain preserving approximation of the Euler equations using convex limiting

A new second-order method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems.

math.NA↗