Backward Error Compensation Algorithms and Their Applications

研究者建议研究前后误差补偿和校正方法（BF）及其应用，特别是用于界面计算、流体动力学和图像处理等方面的水平集方法（Osher和Sethian， 1988）。水平集方程和相关的再距离方程（Sussman et al., 1994）通常由高阶非振荡格式（如ENO、WENO）求解。此外，为了减少界面奇异点附近的扩散，可以使用特殊的技术，如粒子水平集方法（Enright et al., 2002）。BF最初是由Dupont和Liu（2003）开发的，作为解决液位分离时减少扩散的一种简单技术。进一步的改进正在开发中，结果越来越有希望。当应用于Zalesak问题（开槽盘的刚性旋转）时，它接近于流体体积法的分辨率，如Youngs 1982等，并且简单，计算成本低。它还发现了一些特殊的性质，例如当应用于一些不稳定的方案时，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Todd F. Dupont和其他研究人员合作，进一步研究这种算法及其变体和应用。本文研究的具体问题包括：(1)进一步研究高炉的特性，分析不规则网格下高炉的误差和稳定性；(2)基于三角网格的有限元法与水平集法相结合；(3)水平集法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)研究BF在其他微分方程如薛定谔方程中的可能应用。本项目提出的活动涉及计算数学的新方法，并开辟了新的可能性。研究方法将是理论分析与应用相结合。开设适合数学、物理和工程专业学生的综合跨学科课程。本项目开发的新方法将扩大数值方法的范围，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。该项目的进展也将加强几个子领域之间的互动，包括水平集方法，有限元和有限差分方法等。

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.