CFD中高效、精确和易于实现的时间步进方法的一般线性方法设计和optimization
A general linear method approach to the design and optimization of efficient, accurate, and easily implemented time-stepping methods in CFD
摘要：
在流体运动的模拟中，时间精度已被证明是难以捉摸的。我们寻求具有足够强的稳定性的高精度方法来处理许多流的尺度丰富。这些方法还必须易于在当前复杂的（可能是遗留的）代码中实现。在这里，我们开发，分析和测试新的时间步进方法，解决这两个问题的目标是加速时间精确的方法，解决应用程序的需求的发展。新的方法是通过引入廉价的预过滤和后过滤步骤，流行的方法，已实施和测试现有的代码。我们表明，预过滤和后过滤的多步骤或多阶段的方法，在新的方法，其中有多个步骤和阶段的结果：这些是一般的线性方法（GLM）。我们利用GLM的良好研究性质来理解滤波方法的精度和稳定性，并为流行的时间步进方法设计最佳的新滤波器。我们提出了几个新的嵌入式家庭的高精度的方法，具有较低的认知复杂性和良好的稳定性。给出了这些方法的数值试验，包括对某些方法失效点的发现。在新的方法中，提出了一种新的交替过滤器对隐式欧拉法，它导致三阶，A-稳定，误差抑制计划，这是特别有效的。
英文摘要：
In simulations of fluid motion time accuracy has proven to be elusive. We seek highly accurate methods with strong enough stability properties to deal with the richness of scales of many flows. These methods must also be easy to implement within current complex, possibly legacy codes. Herein we develop, analyze and test new time stepping methods addressing these two issues with the goal of accelerating the development of time accurate methods addressing the needs of applications. The new methods are created by introducing inexpensive pre-filtering and post-filtering steps to popular methods which have been implemented and tested within existing codes. We show that pre-filtering and post-filtering a multistep or multi-stage method results in new methods which have both multiple steps and stages: these are general linear methods (GLMs). We utilize the well studied properties of GLMs to understand the accuracy and stability of filtered method, and to design optimal new filters for popular time-stepping methods. We present several new embedded families of high accuracy methods with low cognitive complexity and excellent stability properties. Numerical tests of the methods are presented, including ones finding failure points of some methods. Among the new methods presented is a novel pair of alternating filters for the Implicit Euler method which induces a third order, A-stable, error inhibiting scheme which is shown to be particularly effective.
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