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

Runge-Kutta法(RK)和线性多步法(LM)是求解常微分方程组和偏微分方程组的常用方法。这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方法所能达到的效率，而RK方法在存在刚性和非齐次边界项和源项时精度降低。一般线性(GL)时间步长方法是RK方法和LM方法的推广，因此可以开发出具有更好性能的新的积分格式。然而，GL方法还没有被广泛地研究在依赖于时间的偏微分方程组的背景下，而且几乎没有做什么工作来使这类方法可用于实际应用。这项拟议的研究试图填补这一空白。本研究将从理论上研究一类具有实际意义的一般线性方法的序条件。这一理论将被用来开发新的高阶方法，以避免由于边界、源和刚度而导致的效率和精度降低。对刚性行为的严格分析将在奇异摄动框架下进行，并将扩展到一个微分代数系统的指标。这是首次研究双曲型方程组的强保稳GL格式。将开发一个用于解决多物理问题的分区一般线性格式的框架。新的GL方法将通过一个通用软件包向广大科学界和工程界提供。它们的性能将在预测大气污染时产生的真实、多尺度、多物理模拟中得到说明。

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.