The investigator proposes the study of back and forth error compensation and correction methods (BF) and their applications, in particular, to the level set method (Osher and Sethian, 1988) for interface computation in fluid dynamics and image processing etc. The level set equation and the associated redistancing equation (Sussman et al., 1994) are usually solved by high order non-oscillatory schemes (e.g., ENO, WENO). In addition, special techniques can be used in order to reduce the diffusion near singular 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 set equation. Further improvements are being developed with more and more promising results. When applied to the Zalesak problem (rigid rotation of a slotted disk), it approaches the resolution of volume of fluid methods, e.g., Youngs 1982 etc, and is simple with low computational cost. Some special properties are being found such as that when applied to some unstable schemes, it not only stabilizes them but also improves their accuracy. The investigator plans to collaborate with Todd F. Dupont and other researchers to further study this algorithm and its variants and applications. The particular issues examined in this proposal include: (1)further study of the properties of BF and the error and stability analysis of BF for irregular meshes; (2)the combination of finite element method and level set method with BF on triangular meshes; (3)further development of BF for level set method with applications in fluid dynamics and computer graphics; (4)study of possible applications of BF for other differential equations such as the Schrodinger equation.
The proposed activity in this project involves new methodologies in computational mathematics and opens new possibilities. The research approach will be a combination of theoretical analysis and their applications. An integrated cross-disciplinary curriculum will be developed suitable for students majoring in mathematics, physical sciences and engineering. The new methodologies developed in this project will enlarge the numerical recipes and can be applied to fluid dynamics, interface computation 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. The research results will be disseminated through conference presentations and publications. Progress in this project will also enhance the interaction among several subfields including level set method, finite element and finite difference methods etc.
