Backward Error Compensation Algorithms and Their Applications

研究人员提出了往返误差补偿和校正方法(BF)的研究及其应用，特别是用于界面计算、流体动力学和图像处理等的Level Set方法(Osher和Sethian，1988)，Level Set方程和相关的重新距离方程(Sussman et al.，1994)通常用高阶非振荡格式(如ENO，WENO)求解。此外，可以使用特殊技术来减少界面奇点附近的扩散，例如粒子水平集方法(Enright等人，2002年)。BF最初是由Dupont和Liu(2003)发展起来的，是一种在求解水平集方程时减少扩散的简单方法。正在进行进一步的改进，并取得越来越有希望的结果。当应用于Zalesak问题(狭缝圆盘的刚性旋转)时，它接近于流体体积方法的解，例如Young 1982等，而且计算简单，计算量小。当应用于一些不稳定的格式时，它不仅稳定了它们，而且提高了它们的精度。研究人员计划与托德·F·杜邦和其他研究人员合作，进一步研究该算法及其变体和应用。本文所研究的具体问题包括：(1)进一步研究边界函数的性质以及不规则网格边界函数的误差和稳定性分析；(2)有限元方法和水平集方法与三角网格上边界函数的结合；(3)水平集边界函数的进一步发展及其在流体力学和计算机图形学中的应用；(4)边界函数在其他微分方程如薛定谔方程中的应用研究。研究方法将是理论分析和实际应用的结合。将开发适合数学、物理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配方，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、海底天然气水合物、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报告和出版物传播。该项目的进展还将加强几个子领域之间的相互作用，包括水平集方法、有限元和有限差分法等。

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.