Guido Kanschat Knihy

The book provides an overview of the numerical modelling of radiation fields in multidimensional geometries. It covers advances and problems in the mathematical treatment of the radiative transfer equation, a partial integro-differential equation of high dimension that describes the propagation of the radiation in various fields. Highly accurate and fast methods with sound analytical foundations are particularly emphasized. The volume should not only serve as a basis for the selection of suitable methods for actual radiative transfer problems in disciplines from astronomy and climatology to radiological diagnostics but also be a starting point for additional mathematical investigations on many still unknown aspects of the transfer equation.

This work covers essential topics in numerical analysis and computational methods, focusing on superconvergent techniques. It begins with foundational concepts, including meshes and shape functions, followed by spaces and approximation. The text delves into linear diffusion problems, exploring two distinct sections dedicated to this topic. Subsequent chapters address the Stokes equations and various flow problems, such as the advection-diffusion-reaction equation, Oseen equations, and Navier-Stokes equations. The discussion then shifts to linear solvers, featuring Krylov space methods, interior penalty methods, and local multigrid techniques. It also examines the LDG method and its application to advection-diffusion problems, as well as the Stokes and Oseen equations. Additionally, the book includes example problems that illustrate the application of these methods, covering topics like meshes and domains, the Poisson equation, and the previously mentioned flow equations. Accompanying figures provide visual representations of key concepts, such as eigenvalues on square meshes and the accuracy of various methods. The content is structured to facilitate understanding of complex numerical techniques, making it a valuable resource for those interested in computational fluid dynamics and related fields.