标题： 高阶平衡律系统的方法：基于控制的途径 显示英文标题

标题： High-order well-balanced methods for systems of balance laws: a control-based approach

摘要： 在之前的一些工作中，两位作者介绍了一种策略来开发用于平衡律系统的高阶数值方法，这些方法可以保持系统的所有定态解。这些方法的关键要素是一种平衡重构算子。还引入了一种策略，以修改任何标准的重构算子（如MUSCL、ENO、CWENO等）使其成为平衡的。该策略涉及在每个时间步的每个单元上求解一个非线性问题，即找到一个定态解，其平均值等于给定的单元值。到目前为止，该策略仅应用于那些定态解已知（无论是显式还是隐式形式）的系统。本文的目标是提出一种通用实现这一技术的方法，可应用于任意平衡律系统。为此，在重构过程中需要求解的非线性问题被解释为控制问题：它们包括找到一个ODE系统的解，使得在计算区间上的平均值是给定的。这些问题以泛函形式书写，并基于伴随问题计算泛函的梯度。随后应用牛顿法求解这些问题。特别注意分析使用求积公式计算平均值和源项的影响。为了测试这些方法的有效性和平衡性，将其应用于一系列平衡律系统，从简单的学术系统（如带有某些非线性源项的Burgers方程）到浅水波方程或带有重力效应的气体动力学Euler方程。 显示英文摘要

摘要： In some previous works, two of the authors have introduced a strategy to develop high-order numerical methods for systems of balance laws that preserve all the stationary solutions of the system. The key ingredient of these methods is a well-balanced reconstruction operator. A strategy has been also introduced 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. So far this strategy has been only applied to systems whose stationary solution are known either in explicit or implicit form. The goal of this paper is to present a general implementation of this technique that can be applied to any system of balance laws. To do this, the nonlinear problems to be solved in the reconstruction procedure are interpreted as control problems: they consist in finding a solution of an ODE system whose average at the computation interval is given. These problems are written in functional form and the gradient of the functional is computed on the basis of the adjoint problem. Newton's method is applied then to solve the problems. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. To test their efficiency and well-balancedness, the methods are applied to a number of systems of balance laws, ranging from easy academic systems consisting of Burgers equation with some nonlinear source terms to the shallow water equations or Euler equations of gas dynamics with gravity effects.