论文标题
平衡的适应性紧凑型近似泰勒方法，用于平衡法的系统
Well-balanced adaptive compact approximate Taylor methods for systems of balance laws
论文作者
论文摘要
Carrillo和Pares在2019年引入了紧凑的近似泰勒（CAT）方法。这些方法基于一种策略，该策略允许人们将高级Lax-Wendroff方法扩展到非线性系统，而无需使用Cauchy-Kouchy-Kouchy-Kovalevskaya程序，在使用cauchy-kouchy-kochy-kovalevskaya程序中，均不可分地使用unional interication 2p and inter（2p + 1 porte）（2p + 1点）。最近，在2021年的Carrillo，Macca，Pares，Russo和Zorio提出了一种策略，以摆脱猫方法产生的几乎不连续性的虚假振荡。这种策略导致所谓的自适应猫（ACAT）方法，其中准确性顺序，因此是模板的宽度，适合于溶液的局部平滑度。本文的目的是将CAT和ACAT方法扩展到平衡法律系统。为此，源项写为其无限积分的衍生物，该积分被正式视为通量函数。讨论了该方法的均衡属性，并允许在原则上保留任何固定解决方案的变体。然后，将所得的方法应用于从线性标量保护定律到重力的2D Euler方程的许多系统，并通过带有源项和1D浅水方程的汉堡方程通过：在几个数值测试中检查了阶和平衡的属性。
Compact Approximate Taylor (CAT) methods for systems of conservation laws were introduced by Carrillo and Pares in 2019. These methods, based on a strategy that allows one to extend high-order Lax-Wendroff methods to nonlinear systems without using the Cauchy-Kovalevskaya procedure, have arbitrary even order of accuracy 2p and use (2p + 1)-point stencils, where p is an arbitrary positive integer. More recently in 2021 Carrillo, Macca, Pares, Russo and Zorio introduced a strategy to get rid of the spurious oscillations close to discontinuities produced by CAT methods. This strategy led to the so-called Adaptive CAT (ACAT) methods, in which the order of accuracy, and thus the width of the stencils, is adapted to the local smoothness of the solution. The goal of this paper is to extend CAT and ACAT methods to systems of balance laws. To do this, the source term is written as the derivative of its indefinite integral that is formally treated as a flux function. The well-balanced property of the methods is discussed and a variant that allows in principle to preserve any stationary solution is presented. The resulting methods are then applied to a number of systems going from a linear scalar conservation law to the 2D Euler equations with gravity, passing by the Burgers equations with source term and the 1D shallow water equations: the order and well-balanced properties are checked in several numerical tests.
