Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations
Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations
复制标题
双曲方程的后期/硬松弛渐近保持近似
作者：
C. Berthon;P. LeFloch;R. Turpault
研究了含有刚性松弛项的非线性双曲型守恒律组的解的后期渐近性态。首先，我们引入了Chapman-Enskog型渐近展开式，并导出了描述滞后/刚性松弛奇异极限的有效方程组。讨论了这种新系统的结构，并强调了数学熵的作用。其次，我们提出了一种新的有限体积离散化，在后期的渐近性中，它允许我们恢复相同有效渐近系统的离散版本。只要我们以一种依赖于矩阵值自由参数的方式适当地离散弛豫项，就可以实现这一点，该参数的选择使得期望的渐近行为被获得。我们的结果用几个连续介质物理中感兴趣的模型进行了说明，并且数值实验证明了所提出的理论和数值策略的相关性。
We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems of conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type asymptotic expansion and derive an effective system of equations describing the late-time/stiff relaxation singular limit. The structure of this new system is discussed and the role of a mathematical entropy is emphasized. Second, we propose a new finite volume discretization which, in late-time asymptotics, allows us to recover a discrete version of the same effective asymptotic system. This is achieved provided we suitably discretize the relaxation term in a way that depends on a matrix-valued free-parameter, chosen so that the desired asymptotic behavior is obtained. Our results are illustrated with several models of interest in continuum physics, and numerical experiments demonstrate the relevance of the proposed theory and numerical strategy.