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Collaborative Research: Overcoming Order Reduction and Stability Restrictions in High-Order Time-Stepping

协作研究：克服高阶时间步长中的阶数降低和稳定性限制

基本信息

批准号：
1719637
负责人：
Rodolfo Rosales
金额：
$ 12.33万
依托单位：
Massachusetts Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719637&HistoricalAwards=false
关键词：
Collaborative Research Overcoming Order Reduction

项目摘要

This project develops new computational approaches that remedy fundamental accuracy shortcomings of existing time-stepping methods, and increase their stability and robustness. A wide variety of practical applications, including fluid flows, quantum physics, heat and neutron transport, materials science, and many complex multi-physics problems, require the numerical simulation of models that involve a time evolution. This time evolution must be performed in a way that the high accuracy of modern computational methods is retained. This project addresses fundamental challenges that arise in this context, and delivers superior numerical methods that could replace existing time-stepping schemes currently used in computational science and engineering practice. This project provides a multi-institution collaboration, including two early-career researchers, and it involves the training of a PhD student.The research in this project addresses two aspects in high-order time-stepping: order reduction in Runge-Kutta methods; and unconditionally stable ImEx linear multistep methods. A specific focus lies on time-stepping for partial differential equations. For those, order reduction can be associated with numerical boundary layers, caused by multi-stage time-stepping schemes. Based on this geometric understanding of the phenomenon, remedies for order reduction are developed. This includes the concept of weak stage order, as well as modified boundary conditions. An alternative avenue to avoid order reduction is provided by multistep methods. The key challenge here is their rather restrictive stability behavior. Based on a new stability theory for ImEx multistep methods, this project develops novel schemes that can, for certain problems, achieve unconditional stability. The new schemes can be included into many existing computational codes via a simple modification of the time-stepping coefficients, thus enabling practitioners to select the time step based solely on accuracy considerations.

该项目开发了新的计算方法，可以弥补现有时间步进方法的基本精度缺陷，并提高其稳定性和鲁棒性。各种实际应用，包括流体流动、量子物理、热和中子传输、材料科学以及许多复杂的多物理问题，都需要对涉及时间演化的模型进行数值模拟。这种时间演化必须以保留现代计算方法的高精度的方式进行。该项目解决了在这种情况下出现的基本挑战，并提供了卓越的数值方法，可以取代当前在计算科学和工程实践中使用的现有时间步进方案。该项目提供了多机构合作，包括两名早期职业研究人员，并涉及一名博士生的培训。该项目的研究解决了高阶时间步长的两个方面：龙格-库塔方法的阶数降低；和无条件稳定的 ImEx 线性多步方法。特别关注偏微分方程的时间步长。对于这些，降阶可能与多阶段时间步进方案引起的数值边界层相关。基于对这种现象的几何理解，我们开发了降阶的补救措施。这包括弱阶段顺序的概念以及修改的边界条件。多步骤方法提供了避免订单减少的替代途径。这里的关键挑战是它们相当有限的稳定性行为。基于 ImEx 多步方法的新稳定性理论，该项目开发了新颖的方案，可以针对某些问题实现无条件稳定性。通过对时间步长系数的简单修改，新方案可以被纳入许多现有的计算代码中，从而使从业者能够仅根据精度考虑来选择时间步长。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

A toy model for detonations and flames

爆炸和火焰的玩具模型

DOI：
发表时间：
2017
期刊：
26th International Colloquium on the Dynamics of Explosions and Reactive Systems (ICDERS
影响因子：
0
作者：
Faria, L. M.;Lau-Chapdelaine, S. SM.;Kasimov, A. R.;Rosales, R. R.
通讯作者：
Rosales, R. R.

Unconditional Stability for Multistep ImEx Schemes: Theory