An Enhanced and Highly Efficient Semi-Implicit Combined Lagrange Multiplier Approach Preserving Original Energy Law for Dissipative Systems

IF 2.7 3区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Zhengguang Liu, Nan Zheng, Xiaoli Li
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引用次数: 0

Abstract

Recently, a new Lagrange multiplier approach was introduced by Cheng, Liu, and Shen, which has been broadly used to solve various challenging phase field problems. To design original energy-stable schemes, they have to solve a nonlinear algebraic equation to determine the introduced Lagrange multiplier, which can be computationally expensive, especially for large-scale and long-time simulations involving complex nonlinear terms. In this article, we propose an essential improved technique to modify this issue, which can be seen as a semi-implicit combined Lagrange multiplier approach. In general, the newly constructed schemes keep all the advantages of the Lagrange multiplier method and significantly reduce the computation costs. Besides, the new proposed second-order backward difference formula (BDF2) scheme dissipates the original energy, as opposed to a modified energy for the classic Lagrange multiplier approach. In addition, we establish a general framework for extending our constructed method to dissipative systems. Finally, several examples have been presented to demonstrate the effectiveness of the proposed approach.

一种保留耗散系统原始能量律的增强高效半隐组合拉格朗日乘子法
最近,Cheng, Liu和Shen提出了一种新的拉格朗日乘法器方法,该方法已广泛用于解决各种具有挑战性的相场问题。为了设计原始的能量稳定方案,他们必须求解非线性代数方程来确定引入的拉格朗日乘子,这在计算上是昂贵的,特别是对于涉及复杂非线性项的大规模和长时间模拟。在本文中,我们提出了一种基本的改进技术来修改这个问题,这可以看作是一种半隐式组合拉格朗日乘子方法。总的来说,新构造的方案保留了拉格朗日乘子方法的所有优点,并且大大降低了计算成本。此外,新提出的二阶后向差分公式(BDF2)格式耗散了原始能量,而不是经典拉格朗日乘子方法的修正能量。此外,我们还建立了一个将我们构造的方法推广到耗散系统的一般框架。最后,给出了几个例子来证明所提出方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
5.70
自引率
6.90%
发文量
276
审稿时长
5.3 months
期刊介绍: The International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome.
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