* 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.
Part I. EVOLUTION EQUATIONS IN DISCRETE TIME .
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 .
Methods with Splitting Operator .
Linear Multistep Methods.
Part IV. INTEGRO-DIFFERENTIAL EQUATIONS UNDER DISCRETIZATION .
A Functions of Linear Operators.
B Cauchy Problems in Banach Space.