Description This volume introduces a unified, self-contained study of linear discrete parabolic problems through reducing the starting discrete problem
to the Cauchy problem for an evolution equation in discrete time. Accessible to beginning graduate students, the book contains a general
stability theory of discrete evolution equations in Banach space and gives applications of this theory to the analysis of various classes
of modern discretization methods, among others, Runge-Kutta and linear multistep methods as well as operator splitting methods.
Key
features:
* Presents a unified approach to examining discretization methods for parabolic equations.
* Highlights a stability
theory of discrete evolution equations (discrete semigroups) in Banach space.
* Deals with both autonomous and non-autonomous equations
as well as with equations with memory.
* Offers a series of numerous well-posedness and convergence results for various discretization
methods as applied to abstract parabolic equations; among others, Runge-Kutta and linear multistep methods as well as certain operator
splitting methods.
* Provides comments of results and historical remarks after each chapter.
Audience
Experts and beginners in numerical solution of evolution equations. University professors and teachers.
Contents Preface.
Part I. EVOLUTION EQUATIONS IN DISCRETE TIME .
Preliminaries.
Main Results on Stability.
Operator Splitting
Problems.
Equations with Memory.
Part II. RUNGE-KUTTA METHODS .
Discretization by Runge-Kutta methods .
Analysis of
Stability.
Convergence Estimates .
Variable Stepsize Approximations.
Part III. OTHER DISCRETIZATION METHODS .
The /theta-method.
Methods with Splitting Operator .
Linear Multistep Methods.
Part IV. INTEGRO-DIFFERENTIAL EQUATIONS UNDER DISCRETIZATION .
Integro-Differential Equations.
APPENDIX .
A Functions of Linear Operators.
B Cauchy Problems in Banach Space.
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