Fast Integral Methods for Adaptive Incompressible Flow Simulations

点击翻译按钮获取中文摘要

The proposed research is dedicated to the development and implementation of integral equation methods for linear equations based on the Fast Multipole Method (FMM) for use with adaptive finite difference methods for incompressible flow. The scientific objective of this course of research is to produce new methods for the study of phenomena in incompressible flow systems which are not easily approached with existing methods. For incompressible flow systems, the divergence constraint requires the solution of global elliptic equations which greatly increases the complexity and computational cost of simulation. Many fluid systems also contain a large range of relevant length scales which necessitates the use of spatially and temporally adaptive methods. When applicable, integral equation methods offer the possibility of efficient, adaptive, high-order methods which are readily applicable to complex computational geometries and are also highly parallel. Three distinct applications of FMM will be developed and coupled with adaptive fluid solvers:solution of the Poisson equation, solution of the surface Laplacian,and the fast summation of impulse potentials, each in the context of aspecific physical problem. The numerical study of Boussinesqconvection, in which strong localized fronts develop, requires that astandard Poisson equation be solved. Simulating front formation ofthe quasigeostrophic thermal scalar and internal waves in two fluidsystems requires that a psuedo-differential operator equivalent to asurface Laplacian be inverted. A new approach to modeling thinflexible membranes in incompressible flows requires that membraneforces be evaluated at all grid points as well as a Poisson equationbe solved to enforce incompressibility. For all of these physicalproblems, adaptive methods offer the opportunity to study problemswhich are unapproachable with uniform mesh methods.The range of important fluid flow applications which the proposedresearch could impact is quite diverse. Examples include modeling theocean or climate, predicting oil recovery or contaminant flows in theground, simulating combustion or nuclear reactions, and modeling theflow of blood in organs like the heart or kidneys. For these types ofapplications, the solution of linear equations represents the mostcomputationally intensive part of the overall computer model. The useof integral equation methods to solve the linear equations withinthese models represents a distinct change from the majority of currentcomputer simulations. Any increase in the efficiency in which theseequations can be solved translates directly into the ability forscientists to run larger and more accurate models. The mathematical and computational techniques necessary to make integralequation methods an attractive alternative to more standard approacheshas only been fully developed in the last 10 years. Because of thecomplexity and newness of these methods, little work has been done toexploit their power in applications involving fluid flow. Theapplications that will be pursued as test cases for the new methodsrepresent problems for which current numerical techniques areinadequate for answering fundamental questions of interest toscientists. The algorithms developed will also serve as a steppingstone for future methods applicable to more complicated problems.