High-order numerical methods for hyperbolic conservation laws do not guarantee the validity of constraints that physically meaningful approximations are supposed to satisfy. The finite volume and finite element schemes summarized in this book use limiting techniques to enforce discrete maximum principles and entropy inequalities. Spurious oscillations are prevented using artificial viscosity operators and/or essentially nonoscillatory reconstructions.An introduction to classical nonlinear stabilization approaches is given in the simple context of one-dimensional finite volume discretizations. Subsequent chapters of Part I are focused on recent extensions to continuous and discontinuous Galerkin methods. Many of the algorithms presented in these chapters were developed by the authors and their collaborators. Part II gives a deeper insight into the mathematical theory of property-preserving numerical schemes. It begins with a review of the convergence theory for finite volume methods and ends with analysis of algebraic flux correction schemes for finite elements. In addition to providing ready-to-use algorithms, this text explains the design principles behind such algorithms and shows how to put theory into practice. Although the book is based on lecture notes written for an advanced graduate-level course, it is also aimed at senior researchers who develop and analyze numerical methods for hyperbolic problems.Key Features: oComprehensive review of modern property-preserving schemes for hyperbolic problemsoUnification of algorithms developed for finite volume and finite element approximationsoNew-generation limiters for finite element methods and nonlinear hyperbolic systemsoSubcell flux correction tools for arbitrarily high-order discretizations in space and timeoMany new algorithms including those that have not been published previouslyoIntroduction to the theoretical foundations of algebraic flux correction schemesoA toolbox for and a guide to developing new advanced discretization methods

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