Numerical methods for kinetic equations*

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2014-05-01 DOI:10.1017/S0962492914000063

G. Dimarco, L. Pareschi

引用次数: 266

Abstract

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

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动力学方程的数值方法*

在这篇综述中，我们考虑了动力学偏微分方程数值方法的发展和数学分析。动力学方程是描述由大量粒子组成的系统的时间演化的一种方法。由于高维数及其固有的物理性质，数值方法的构建是一个挑战，需要在精度和计算复杂性之间取得谨慎的平衡。这里我们回顾了处理这类方程的基本数值技术，包括半拉格朗日方法、离散速度模型和谱方法。此外，我们还概述了动力学方程的数值方法的现状。这包括快速算法的推导，渐近保持方法的概念和混合格式的构造。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.