Purpose

The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.

Design/methodology/approach

The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.

Findings

The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.

Originality/value

To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

中文翻译：

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关于通过通用 A-后验误差估计器将 GS4-1 框架应用于流体动力学和自适应时间步长

目的

本文的目的是描述一种新的通用误差估计器和广义单步单解 (GS4-1) 计算框架中的自适应时间步长过程，应用于流体动力学，并附有不可压缩 Navier-Stokes 方程的图解.

设计/方法/方法

所提出的误差估计器是通用且通用的，它适用于 GS4-1 框架的整个子集，包括非耗散 Crank-Nicolson 方法、最耗散的反向微分公式以及介于两者之间的任何方法。可以避免针对特定时间积分算法的繁琐的误差估计设计工作，具有新颖性和新颖性。在数值实现上，局部误差估计有一个紧凑的表示，它由四个连续时间级别的时间导数变量确定，并且只涉及向量运算，这对于数值实现来说是简单的。此外，所提出的误差估计器进一步说明了自适应时间步长，并用于解决盖子驱动型腔和流过圆柱体的基准问题。

发现

所提出的计算程序能够消除 GS4-1(1,1)/Crank-Nicolson 方法中的非物理振荡；在耗散和非耗散方案中均具有 CPU 效率，具有更好的解算精度；并检测复杂的物理场，从而根据用户定义的误差阈值选择合适的时间步长。

原创性/价值

据作者所知，本研究首次将通用 GS4-1 系列时间积分算法应用于流体动力学中具有恒定和自适应时间步长的不可压缩 Navier-Stokes 方程的瞬态模拟通用误差估计器。所提出的计算框架对于数值实现来说很简单，并且基于所提出的误差估计的时间步长选择是有效的，有利于瞬态模拟的计算开销。