CIF:Small: General Linear Time-stepping Methods for Large-Scale Simulations

CIF:Small：用于大规模仿真的通用线性时间步进方法

• 批准号: 0916493
• 负责人: Adrian Sandu
• 金额: $ 31.23万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0916493&HistoricalAwards=false
• 关键词: CIF; Small; General; Linear; Time

General Linear Time-stepping Methods for Large Scale Simulations Runge-Kutta(RK) and linear multistep (LM) methods have been extensively used for the integration of ordinary and partial differential equations (PDEs). Both families of methods have well known limitations. Stability requirements limit the efficiency attainable by any LM method, whereas RK methods suffer from accuracy reduction in the presence of stiffness and nonhomogeneous boundary and source terms. General linear (GL) time-stepping methods are generalizations of both RK and LM methods and therefore allow the development of new integration schemes with superior properties. However, GL methods have not been extensively studied in the context of time-dependent PDEs, and very little has been done to make this class of methods available for practical use. The proposed research seeks to fill this gap. This research will investigate theoretically order conditions for a class of general linear methods of practical importance. This theory will be used to develop new high order methods that circumvent the efficiency and accuracy reduction due to boundaries, sources, and stiffness. A rigorous analysis of the stiff behavior will be carried out in a singular perturbation framework, and will be extended to index one differential algebraic systems. The proposed research is the first to address strong stability preserving GL schemes for hyperbolic systems. A framework for partitioned general linear schemes will be developed to address multiphysics problems. The new GL methods will be made available to the science and engineering community at large through a general purpose software package. Their performance will be illustrated on real life, multiscale, multiphysics simulations arising in the prediction of atmospheric pollution.

大规模仿真的通用线性时间步长方法Runge-Kutta（RK）和线性多步（LM）方法已广泛用于整合普通和部分微分方程（PDES）。两种方法都有众所周知的局限性。稳定性要求限制了任何LM方法可实现的效率，而RK方法在刚度和非均匀边界和源术语的存在下的准确性降低。一般线性（GL）时间步变方法是RK和LM方法的概括，因此允许开发具有出色特性的新集成方案。但是，GL方法尚未在时间依赖性PDE的背景下进行广泛研究，并且几乎没有采取很少的研究来使这类方法可用于实际使用。拟议的研究旨在填补这一空白。这项研究将研究理论上订购一类一类实用重要性线性方法的条件。该理论将用于开发新的高级方法，以规避由于边界，源和刚度降低效率和准确性。对僵硬行为的严格分析将在一个奇异的扰动框架中进行，并将扩展到索引一个差分代数系统。拟议的研究是第一个解决强大稳定性保留双曲线系统的稳定性的研究。将开发用于分区的一般线性方案的框架来解决多物理问题。新的GL方法将通过通用软件包为科学和工程社区提供。他们的表现将在现实生活，多尺度，多物理模拟中进行说明，这些模拟在预测大气污染中。