关键词:
Coordinate transformations
Algorithms
Turbulence models
Flow velocity
Vortices
Fluid dynamics
Reynolds number
Stencils
Computer science
Fluid mechanics
Mechanics
Physics
摘要:
We study in this thesis the Characteristic Mapping (CM) framework and its application in various scientific computing problems involving transported quantities. The CM framework is a novel numerical approach for general transport equations; it consists in discretizing the deformation map generated by the transport flow instead of the transported quantities themselves. This computation yields the solution operator of the advection equation, called the characteristic map, which can then be used to transport any quantity evolving under the same transport flow. The CM method is distinguished by several unique numerical properties made possible by its geometric approach: any number of transported quantities can be simultaneously and coherently evolved using the same characteristic map, the evaluation of advected fields through map pullbacks guarantees relabelling symmetry and thus eliminates artificial viscous dissipation, the group structure of the characteristic maps can also be used to decompose long-time maps into a number of short-time submaps allowing for efficient coarse grid computations. The resulting numerical method is efficient, accurate and well-suited for multiscale problems such as the simulation of incompressible inviscid fluids. We will first motivate the CM approach from the linear advection equation. We will also study the some underlying theory and preliminaries on which the CM method is built, in particular, we will examine relevant differential calculus concepts which allow us to formulate the characteristic map as the solution operator to the Lie-advection equation. As a first application, we study the use of the CM method for the problem of equiareal surface parametrization in the context of passively advected surfaces. In this case, the CM method is used to produce a redistribution map on the parametric space which, when composed with the evolving parametrization, yields an equiareal parametrization of the surface. Next, we apply the CM method to