Physics motivated algorithms for partial differential equations.

摘要

Many nonequilibrium phenomena are spatially extended, and the most popular means to model them is the partial differential equation (PDE). Resultant PDEs are, however, often nonlinear, defying analytical approaches. Thus, devising efficient numerical algorithms to solve PDEs is important for the study of nonequilibrium systems, and has traditionally been considered a major branch of applied mathematics.; A computationally efficient model that captures the crucial physics of a system can be an efficient numerical solver of the PDE describing the system. This general idea will be illustrated in terms of solvers for hyperbolic equations, such as those describing advection in fluids and linear wave propagation. We demonstrate in this thesis that a conscious pursuit of physics essence can lead to useful numerical algorithms. From this point of view, the development of solvers for physically meaningful PDEs can be considered a branch of applied physics.; Our strategy for deriving new algorithms is to implement the crucial physics, as faithfully as possible, in order to reproduce the phenomenon inside the computer. The solution of the PDE is obtained, in this approach as a by-product of the correct implementation of the physics of the problem. After explaining the derivation of algorithms for the solution of advection in fluids, we present a new methodology to derive algorithms for wave propagation problems, based on the modelling of Huygens' principle. The new methodology can be used to derive higher-order algorithms systematically. We explain why these algorithms are advantageous in comparison to standard higher-order finite-difference algorithms, and present tests and evaluations of the new schemes. We give new algorithms for the wave equation and Maxwell's equations, including the implementation of some types of boundary conditions. We conclude by suggesting extensions of the method to related problems.