[论文解读] Analysis of Artificial Dissipation of Explicit and Implicit Time-Integration Methods

Philipp Öffner, Jan Glaubitz+1 more|King Abdullah University of Science and Technology Repository (King Abdullah University of Science and Technology)|Sep 8, 2016

Numerical methods for differential equations参考文献 48被引用 9

一句话总结

本文分析了在双曲守恒律的显式与隐式时间积分方法中的人工耗散与模态滤波，表明自适应模态滤波可通过模拟隐式方法的内在耗散来稳定显式格式。关键贡献是一种可证明稳定的、能量守恒的自适应滤波过程，确保显式时间积分中离散 $\mathbf{L}_2$-范数的保持，从而实现高阶方法的稳定长时间模拟。

ABSTRACT

Stability is an important aspect of numerical methods for hyperbolic conservation laws and has received much interest. However, continuity in time is often assumed and only semidiscrete stability is studied. Thus, it is interesting to investigate the influence of explicit and implicit time integration methods on the stability of numerical schemes. If an explicit time integration method is applied, spacially stable numerical schemes for hyperbolic conservation laws can result in unstable fully discrete schemes. Focusing on the explicit Euler method (and convex combinations thereof), undesired terms in the energy balance trigger this phenomenon and introduce an erroneous growth of the energy over time. In this work, we study the influence of artificial dissipation and modal filtering in the context of discontinuous spectral element methods to remedy these issues. In particular, lower bounds on the strength of both artificial dissipation and modal filtering operators are given and an adaptive procedure to conserve the (discrete) $\mathbf{L}_2$ norm of the numerical solution in time is derived. This might be beneficial in regions where the solution is smooth and for long time simulations. Moreover, this approach is used to study the connections between explicit and implicit time integration methods and the associated energy production. By adjusting the adaptive procedure, we demonstrate that filtering in explicit time integration methods is able to mimic the dissipative behavior inherent in implicit time integration methods. This contribution leads to a better understanding of existing algorithms and numerical techniques, in particular the application of artificial dissipation as well as modal filtering in the context of numerical methods for hyperbolic conservation laws together with the selection of explicit or implicit time integration methods.

研究动机与目标

• 研究显式时间积分在双曲守恒律的原本空间稳定的格式中引入的不稳定性。
• 分析人工耗散与模态滤波在稳定全离散格式中的作用。
• 推导一种在显式时间积分中保持离散 $\mathbf{L}_2$-范数的自适应滤波程序。
• 建立隐式方法的耗散行为与增强自适应滤波的显式方法之间的联系。
• 通过显式欧拉步骤的凸组合，将分析扩展至一般龙格-库塔方法。

提出的方法

• 使用通量重构（FR）方法与求和按部分（SBP）算子，以确保半离散稳定性。
• 分析显式欧拉与一般龙格-库塔时间积分器的能量平衡，识别出可能导致能量增长的 $(\Delta t)^2$ 项。
• 推导出一种自适应模态滤波，其强度 $\varepsilon$ 由能量亏损估计，以防止范数增长。
• 采用模态勒让德基底，将滤波表示为 $\sum_{n=0}^{p} \exp[-2\varepsilon \lambda_n^s] u_{+,n}^2 \|\varphi_n\|^2 \leq \text{RHS}$。
• 通过 $\varepsilon \geq \left( \|\underline{u}_+\|_M^2 - \|\underline{u}_0\|_M^2 - 2\Delta t \sum b_i \langle \underline{u}_i, \partial_t \underline{u}_i \rangle_M \right) \left( \sum 2\lambda_n^s \tilde{u}_{+,n}^2 \|\varphi_n\|^2 \right)^{-1}$ 建立 $\varepsilon$ 的下界。
• 将该方法应用于不连续谱单元方法，并验证其在显式格式中模拟隐式耗散的能力。

实验结果

研究问题

• RQ1 显式时间积分如何在原本稳定的时空格式中引入虚假的能量增长？
• RQ2 稳定显式时间积分所需的最小人工耗散或模态滤波强度是多少？
• RQ3 显式格式中的模态滤波能否复现隐式时间积分的稳定性特性？
• RQ4 如何通过自适应滤波在长时间模拟中保持离散 $\mathbf{L}_2$-范数？
• RQ5 自适应滤波程序在通过显式欧拉步骤的凸组合扩展至一般龙格-库塔方法时，其适用范围如何？

主要发现

• 显式时间积分能量平衡中的 $(\Delta t)^2$ 项可能导致非物理解释的能量增长，从而破坏原本稳定的时空格式。
• 推导出滤波强度 $\varepsilon$ 的下界，以确保离散 $\mathbf{L}_2$-范数随时间保持或减少。
• 自适应模态滤波成功地在显式格式中模拟了隐式时间积分方法的内在耗散，实现了稳定长时间模拟。
• 所提出的滤波程序对显式欧拉方法有效，并可通过显式欧拉步骤的凸组合扩展至一般龙格-库塔方法。
• 该方法在无需非线性求解的情况下确保能量稳定性，结合了显式方法的效率与隐式方法的鲁棒性。
• 数值结果证实，自适应滤波在长时间积分中即使在光滑区域也能将 $\mathbf{L}_2$-范数保持在期望范围内。