Nonstandard High Order Multigrid Techniques with Applications to Laminar Diffusion Flame Simulations

本计画将设计有效的几何多重网格方法，利用中间网格来解决由高阶紧致格式离散的对流扩散问题。使用中间网格的目的是减少在不同网格上得到的解之间的差异。符号计算包将用于推导中间网格上的四阶有限差分格式。将制定特殊的放宽方案和并网转移运营商。层流扩散火焰的数值模拟将采用高阶离散化格式的非标准多重网格法。中间网格的使用有望缓解现有多网格方法在火焰模拟中存在的问题，即收敛的粗网格校正不在细网格牛顿迭代的收敛域内。这个项目需要符号和数值计算技术。一个有用的符号计算程序的成功开发将极大地促进数值计算界对符号计算包的认识和使用。本课题的研究成果将对对流扩散问题的几何多网格解的理解以及高阶紧致格式在实际流动模拟中的应用做出重要贡献。这类问题的有效求解是计算流体力学中许多数值模拟的核心。快速层流扩散火焰代码在燃烧和环境保护方面具有重要意义。与商业燃烧器、污染物追踪、汽车和飞机制造、燃烧相关的某些美国行业可以从这项研究中受益。

This project will design efficient geometric multigrid methods using intermediate grids to solve convection diffusion problems discretized by high order compact schemes. The use of intermediate grids aims at reducing discrepancy between the solutions obtained on differentgrids. Symbolic computation packages will be used to derive a fourth order finite difference scheme on the intermediate grids. Special relaxation schemes and intergrid transfer operators will be developed.The nonstandard multigrid method with high order discretization schemes will be used in the numerical simulation of laminar diffusion flames. The use of intermediate grids is expected to alleviate the problem haunting existing multigrid methods in the flame simulation, in which the converged coarse grid correction is not in the convergence domain of the fine grid Newton iteration.This project requires both symbolic and numerical computation techniques. The successful development of a useful symbolic computation procedure will greatly promote the awareness and use of symbolic computation packages in numerical computation community.The results of this research project will make important contribution to the understanding of geometric multigrid solution of convectiondiffusion problems, and of the applications of high order compactschemes to realistic flow simulations. Efficient solution of such problems is central to many numerical simulations in computational fluid dynamics. The fast laminar diffusion flame code is useful in combustion and environment protection. Certain U.S. industries related to commercial burners, pollutant tracking, car and airplane manufacturing, combustion, can benefit from this research.