Numerical Solution of Differential Equations

The material that this book covers can be viewed as a first course on the numerical solution of differential equations. It is designed for people who want to gain a practical knowledge of the techniques used today. The course aims to achieve a thorough understanding of the issues and methods involved and of the reasons for the successes and failures of existing software.

The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, in both theory and applications, emphasising the numerical methods needed to solve such equations. It assumes of the reader an undergraduate background in mathematical methods typical of engineers and physicists, though many chapters begin with a descriptive summary.

Domain decomposition methods are divide and conquer methods for the parallel and computational solution of partial differential equations of elliptic or parabolic type. They include iterative algorithms for solving the discretized equations, techniques for non-matching grid discretizations and techniques for heterogeneous approximations.