Positivity-Preserving Lax–Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models
Positivity-Preserving Lax–Wendroff Discontinuous Galerkin Schemes for Quadrature-Based Moment-Closure Approximations of Kinetic Models
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DOI：
10.1007/s10915-023-02117-5
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发表时间：
2021-11
影响因子：
2.5
通讯作者：
Erica R. Johnson;J. Rossmanith;Christine Vaughan
中科院分区：
文献类型：
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作者：
Erica R. Johnson;J. Rossmanith;Christine Vaughan
基于矩量法（QRST）提供了一类很有前途的近似技术，用于将动力学方程简化为在热力学平衡之外有效的流体方程。在这项工作中，我们研究了一个特定的五个时刻的变体称为HyQeroes的Qeroes，并建立该系统是在解空间的凸区域的时刻可逆。然后，我们开发了一个高阶不连续Galerkin（DG）计划解决由此产生的流体系统。该计划是基于预测-校正方法，其中的预测是一个本地化的空间-时间DG计划。该预测中的非线性代数系统使用Picard迭代求解。修正是基于演化方程的时间积分的直接显式更新，其中时空预测取代了精确解的所有实例。在没有限制器的情况下，高阶方案并不能保证解保持在HyQdom矩可实现的凸集上。为了克服这一点，我们引入了新的限制器，严格保证计算的解决方案不会离开可实现的解决方案的凸集，从而保证系统的双曲性。我们在预测和校正步骤中开发了保正限制器，并开发了一个振荡限制器，用于抑制冲击附近的非物理振荡。我们还开发了一种新的扩展，该计划包括BGK碰撞算子，所提出的方法被证明是渐近保持在高碰撞极限。的HyQNOW和HyQMOM-BGK求解器进行了验证，在几个测试用例，表现出高阶精度光滑的问题和冲击捕捉能力的问题与冲击。数值验证了HyQMOM-BGK求解器的渐近保持特性。
The quadrature-based method of moments (QMOM) offers a promising class of approximation techniques for reducing kinetic equations to fluid equations that are valid beyond thermodynamic equilibrium. In this work, we study a particular five-moment variant of QMOM known as HyQMOM and establish that this system is moment-invertible over a convex region in solution space. We then develop a high-order discontinuous Galerkin (DG) scheme for solving the resulting fluid system. The scheme is based on a predictor–corrector approach, where the prediction is a localized space-time DG scheme. The nonlinear algebraic system in this prediction is solved using a Picard iteration. The correction is a straightforward explicit update based on the time-integral of the evolution equation, where the space-time prediction replaces all instances of the exact solution. In the absence of limiters, the high-order scheme does not guarantee that solutions remain in the convex set over which HyQMOM is moment-realizable. To overcome this, we introduce novel limiters that rigorously guarantee that the computed solution does not leave the convex set of realizable solutions, thus guaranteeing the hyperbolicity of the system. We develop positivity-preserving limiters in both the prediction and correction steps and an oscillation limiter that damps unphysical oscillations near shocks. We also develop a novel extension of this scheme to include a BGK collision operator; the proposed method is shown to be asymptotic-preserving in the high-collision limit. The HyQMOM and the HyQMOM-BGK solvers are verified on several test cases, demonstrating high-order accuracy on smooth problems and shock-capturing capability on problems with shocks. The asymptotic-preserving property of the HyQMOM-BGK solver is also numerically verified.