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

Runge-Kutta（RK）和线性多步（LM）方法已广泛用于常微分方程和偏微分方程的积分。这两类方法都有众所周知的局限性。稳定性要求限制了任何LM方法可达到的效率，而RK方法在刚度和非齐次边界和源项的存在下精度降低。一般线性时间步方法是RK和LM方法的推广，因此可以发展具有上级性质的新的积分格式。然而，GL方法还没有得到广泛的研究，在时间相关的偏微分方程的背景下，很少已经做了这类方法可用于实际应用。这项研究旨在填补这一空白。本文将从理论上研究一类具有实际意义的一般线性方法的阶条件。这一理论将被用来开发新的高阶方法，规避由于边界，源和刚度的效率和精度降低。在奇异摄动框架下对刚性行为进行了严格的分析，并将其扩展到指数为1的微分代数系统。所提出的研究是第一个解决双曲型系统的强稳定性保持GL格式。一个框架划分一般线性计划将开发解决多物理场问题。新的地球引力方法将通过一个通用软件包提供给广大的科学和工程界。它们的性能将在大气污染预测中出现的真实的生活、多尺度、多物理场模拟中得到说明。

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.