用过滤对流速度正则化Burgers方程
A Regularization of Burgers Equation using a Filtered Convective Velocity
摘要：
本文研究了一维和多维Burgers方程的正则化问题，我们称之为对流过滤Burgers（CFB）方程。过滤技术背后的物理动机。证明了一个多维和一般类滤子的存在唯一性定理。多个不变量的CFB方程的运动发现，并与粘性和无粘Burgers方程中发现的。对于一般的滤波器类，得到了行波解，并证明了其收敛于具有正确波速的无粘Burgers方程的弱解.精确的数值模拟进行了一维和二维的情况下，激波的行为，激波厚度，和动能衰减进行了检查。能量谱也被检查，并显示出与光滑的解决方案。
英文摘要：
This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation and are compared with those found in viscous and inviscid Burgers equation. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of inviscid Burgers equation with the correct wave speed. Accurate numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness, and kinetic energy decay are examined. Energy spectrum are also examined and are shown to be related to the smoothness of the solutions.
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