Backward Error Compensation Algorithms and Their Applications

后向误差补偿算法及其应用

• 批准号: 0511815
• 负责人: Yingjie Liu
• 金额: $ 10.9万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0511815&HistoricalAwards=false
• 关键词: Backward; Error; Compensation; Algorithms; Their

The investigator proposes the study of back and forth error compensationand correction methods (BF) and their applications, in particular, to thelevel set method (Osher and Sethian, 1988) for interface computation influid dynamics and image processing etc. The level set equation and theassociated redistancing equation (Sussman et al., 1994) are usually solvedby high order non-oscillatory schemes (e.g., ENO, WENO). In addition,special techniques can be used in order to reduce the diffusion nearsingular points of the interface, such as the particle level set method(Enright et al., 2002). BF was initially developed by Dupont and Liu (2003)as a simple technique for reducing the diffusion in solving the level setequation. Further improvements are being developed with more and morepromising results. When applied to the Zalesak problem (rigid rotationof a slotted disk), it approaches the resolution of volume of fluidmethods, e.g., Youngs 1982 etc, and is simple with low computational cost.Some special properties are being found such as that when applied to someunstable schemes, it not only stabilizes them but also improves theiraccuracy. The investigator plans to collaborate with Todd F. Dupont andother researchers to further study this algorithm and its variants andapplications. The particular issues examined in this proposal include:(1)further study of the properties of BF and the error and stabilityanalysis of BF for irregular meshes; (2)the combination of finite elementmethod and level set method with BF on triangular meshes; (3)furtherdevelopment of BF for level set method with applications in fluid dynamicsand computer graphics; (4)study of possible applications of BF for otherdifferential equations such as the Schrodinger equation.The proposed activity in this project involves new methodologies incomputational mathematics and opens new possibilities. The researchapproach will be a combination of theoretical analysis and theirapplications. An integrated cross-disciplinary curriculum will be developedsuitable for students majoring in mathematics, physical sciences andengineering. The new methodologies developed in this project will enlargethe numerical recipes and can be applied to fluid dynamics, interfacecomputation and their applications like atmospheric dynamics, ocean flow,ocean floor gas hydrate, crystal growth, biological fluid dynamics, elastic-plastic solids, astronomy, computer graphic, image processing, etc. Theresearch results will be disseminated through conference presentationsand publications. Progress in this project will also enhance theinteraction among several subfields including level set method, finiteelement and finite difference methods etc.

本文提出了一种新的误差补偿和校正方法（BF）及其应用，特别是在流体力学和图象处理等领域中界面计算的水平集方法（Osher和Sethian，1988）中的应用。1994）通常由高阶非振荡格式（例如，ENO，韦诺）。此外，可以使用特殊技术以减少界面的扩散接近奇点，例如粒子水平集方法（Enright等人，2002年）。BF最初是由Dupont和Liu（2003）开发的，作为在求解能级方程时减少扩散的简单技术。进一步的改进正在开发，越来越有希望的结果。当应用于Zalesak问题（开槽圆盘的刚性旋转）时，它接近流体体积方法的分辨率，例如，Youngs（1982）等提出的新格式，它具有简单、计算量小的特点，并发现了一些特殊的性质，如当应用于某些不稳定格式时，它不仅稳定了这些格式，而且提高了它们的精度。调查人员计划与托德F. Dupont等研究人员进一步研究了该算法及其变体和应用.本文的研究内容包括：（1）进一步研究BF的性质，以及BF在不规则网格上的误差和稳定性分析;（2）将有限元法和水平集方法与BF在三角形网格上相结合;（3）进一步发展水平集方法的BF，并将其应用于流体力学和计算机图形学;（四）研究BF在其他微分方程（如薛定谔方程）中的可能应用。该项目的建议活动涉及计算数学的新方法开启了新的可能性研究方法将是理论分析与实际应用相结合。将开发适合数学、物理科学和工程专业学生的综合跨学科课程。该项目开发的新方法将扩大数值方法，并可应用于流体动力学、界面计算及其应用，如大气动力学、海洋流动、海底气体水合物、晶体生长、生物流体动力学、弹塑性固体、天文学、计算机图形学、图像处理等。研究成果将通过会议演示和出版物传播。该项目的进展也将促进水平集方法、有限元和有限差分方法等几个子领域之间的相互作用。