Numerical methods of ordinary differential equations. Initial and boundary value problems (Q5891794)

The book represents a useful introduction to standard numerical solution methods for initial and boundary value problems in ordinary differential equations. Starting from elementary modeling aspects, numerical methods are introduced for initial value problems, ranging from Taylor series methods to Runge-Kutta methods and linear multistep methods, covering the classical stability and convergence theory and practical aspects of adaptive step-size selection. Stiff problems are also discussed, where the attempt at a characterization of this problem class does not appear uncontroversial, particularly for nonlinear problems. The presentation of discontinuous Galerkin methods for initial value problems seems a bit unusual for this problem class in the framework of a textbook, but demonstrates that error bounds for these methods require weaker regularity assumptions than the classical methods.NEWLINENEWLINE For boundary value problems, the first presented solution approach are shooting methods. The drawbacks associated with dichotomy and instability are not elaborated in sufficient detail such as to yield motivation for global methods such as difference schemes, collocation or Galerkin methods. As theoretical tools in the error analysis, maximum principles and compactness arguments are employed. While these represent very useful approaches in general which require low solution regularity, for collocation the well-known (super-)convergence orders cannot be reproduced, whence a very important aspect of high-order methods is not reflected.NEWLINENEWLINE In summary, the author gives a useful overview over wide classes of numerical methods for the solution of initial and boundary value problems for ordinary differential equations, which inevitably cannot be comprehensive and lacks aspects which may be considered important by some researchers, but also discusses finite element Galerkin methods which represents a useful supplement to many standard textbooks on the subject.NEWLINENEWLINEFor the first edition see [de Gruyter Lehrbuch. Berlin: de Gruyter (2008; Zbl 1144.65048)].