Exponentially Convergent Algorithms for Abstract Differential Equations

Suitable for practical computing, this volume presents accurate and efficient exponentially convergent methods for abstract, differential equations. Each equation can be viewed as a meta-model of systems of both ordinary and partial differential equations.

This book presents new accurate and efficient exponentially convergent methods for abstract differential equations with unbounded operator coefficients in Banach space. These methods are highly relevant for practical scientific computing since the equations under consideration can be seen as the meta-models of systems of ordinary differential equations (ODE) as well as of partial differential equations (PDEs) describing various applied problems. The framework of functional analysis allows one to obtain very general but at the same time transparent algorithms and mathematical results which then can be applied to mathematical models of the real world. The problem class includes initial value problems (IVP) for first order differential equations with constant and variable unbounded operator coefficients in a Banach space (the heat equation is a simple example), boundary value problems for the second order elliptic differential equation with an operator coefficient (e.g. the Laplace equation), IVPs for the second order strongly damped differential equation as well as exponentially convergent methods to IVPs for the first order nonlinear differential equation with unbounded operator coefficients. For researchers and students of numerical functional analysis, engineering and other sciences this book provides highly efficient algorithms for the numerical solution of differential equations and applied problems.

Contains the first comprehensive and unified theory of exponentially convergent numerical methods for the differential equations with operator coefficients including many important PDEs as particular cases Algorithms inherit a two-level parallelism with respect to both the spatial and the time variables All other books deal with polynomially convergent methods such as the finite difference methods, finite element methods etc.

Contenu

Preface.- 1 Introduction.- 2 Preliminaries.- 3 The first-order equations.- 4 The second-order equations.- Appendix: Tensor-product approximations of the operator exponential.- Bibliography.- Index.