A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws
A self-adaptive theta scheme using discontinuity aware quadrature for solving conservation laws
复制标题
使用不连续感知求积求解守恒定律的自适应 theta 方案
DOI：
10.1093/imanum/drab071
复制
发表时间：
2021
影响因子：
2.1
通讯作者：
Huang, Chieh-Sen
中科院分区：
文献类型：
--
作者：
Arbogast, Todd;Huang, Chieh-Sen
本文提出了一个间断感知求积（DAQ）规则，并利用它发展了一个隐式自适应θ（SATh）格式来逼近标量双曲型守恒律。我们的SATH计划需要解决两个方程的系统，一个控制在时间水平上的解决方案的细胞平均值，和其他控制的解决方案的时空平均值。这些量在DAQ规则中用于精确地近似双曲线通量函数的时间积分，即使在时间间隔上的某个地方解可能是不连续的。其结果是一个有限体积计划，使用θ时间步进方法，θ定义隐式（或自适应）。两个计划的开发，自适应θ上游加权（SATh-up）的单调通量函数使用简单的上游稳定，和自适应θ Lax-Friedrichs（SATh-LF）使用Lax-Friedrichs数值通量。我们证明了DAQ是精确到二阶时，有一个不连续的解决方案和三阶时，它是光滑的。我们证明了SATh-up是无条件稳定的，只要theta被设置为至少1/2（这意味着SATh通常只能是一阶精度）。我们还证明了SATh-up满足最大值原理，并且在适当的单调性和边界条件下是总变差递减的。一般的通量函数需要SATh-LF计划，所以我们评估其精度通过数值例子在一个和两个空间维度。这些结果表明SATh-LF也是稳定的，并且满足最大值原理（至少在合理的Courant-Friedrichs-Lewy数下）。与采用Crank-Nicolson和向后Euler时间推进的有限体积格式的解相比，SATh-LF解的精度往往接近前者，但没有振荡，数值扩散性也比后者小。
We present adiscontinuity aware quadrature(DAQ) rule and use it to develop implicitself-adaptive theta(SATh) schemes for the approximation of scalar hyperbolic conservation laws. Our SATh schemes require the solution of a system of two equations, one controlling the cell averages of the solution at the time levels, and the other controlling the space-time averages of the solution. These quantities are used within the DAQ rule to approximate the time integral of the hyperbolic flux function accurately, even when the solution may be discontinuous somewhere over the time interval. The result is a finite volume scheme using the theta time stepping method, with theta defined implicitly (or self-adaptively). Two schemes are developed, self-adaptive theta upstream weighted (SATh-up) for a monotone flux function using simple upstream stabilization, and self-adaptive theta Lax–Friedrichs (SATh-LF) using the Lax–Friedrichs numerical flux. We prove that DAQ is accurate to second order when there is a discontinuity in the solution and third order when it is smooth. We prove that SATh-up is unconditionally stable, provided that theta is set to be at least 1/2 (which means that SATh can be only first order accurate in general). We also prove that SATh-up satisfies the maximum principle, and is total variation diminishing under appropriate monotonicity and boundary conditions. General flux functions require the SATh-LF scheme, so we assess its accuracy through numerical examples in one and two space dimensions. These results suggest that SATh-LF is also stable and satisfies the maximum principle (at least at reasonable Courant-Friedrichs-Lewy numbers). Compared to the solutions of finite volume schemes using Crank–Nicolson and backward Euler time stepping, SATh-LF solutions often approach the accuracy of the former, but without oscillation, and they are numerically less diffuse than the latter.