标题： 配置方法对于高阶平衡法在平衡律系统中的应用 显示英文标题

标题： Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

摘要： 在一些先前的工作中，两位作者介绍了一种设计一维守恒律高阶数值方法的技术，这些方法可以保持它们的所有定态解。这些方法的基础是一种平衡重构算子。此外，他们还介绍了一种修改任何标准重构算子（如MUSCL、ENO、CWENO等）的方法，使其成为平衡的。该策略涉及在每个时间步的每个单元中求解一个非线性问题，即找到一个定态解，使得其平均值等于给定的单元值。在最近的一篇论文中，提出了一种完全平衡的方法，其中重构过程中需要求解的非线性问题被解释为控制问题。本文的目标是引入一种基于配置RK方法来解决这些局部非线性问题的新技术。特别注意分析使用求积公式计算平均值和源项的影响。还介绍了一种通用技术，以处理共振问题。为了验证方法的有效性和平衡特性，已将其应用于一系列测试，从简单的具有非线性源项的Burgers方程到浅水方程（带或不带 Manning 摩擦）以及带有重力效应的欧拉气体动力学方程。 显示英文摘要

摘要： In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations -- with and without Manning friction -- or Euler equations of gas dynamics with gravity effects.