Backward Error Compensation Algorithms and Their Applications

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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.