About the book

    This book presents numerical methods based on operator splitting for a class of nonlinear partial differential equations possessing solutions with limited regularity or even discontinuous solutions. This class is exemplified by nonlinear and strongly degenerate convection-diffusion equations in several spatial dimensions, which in turn encompass, e.g., hyperbolic conservation laws, nonlinear diffusion (porous medium) equations, and the Buckley-Leverett equation. 

    The book describes, in a self-contained manner, robust and efficient numerical methods that also yield mathematical insight. The numerical methods are shown to converge to the exact solution, thereby offering constructive proofs of existence of properly defined solutions of these equations. A cornerstone of the numerical approach is the use of operator splitting, where complicated problems are split into simpler problems, each treated separately. The simpler problems are dealt with using, e.g., finite difference schemes and front tracking. 

    A notable part of this book reports the results of applying operator splitting methods to a variety of convection dominated problems; this includes problems coming from applications like flow in porous media, shallow water waves, and gas dynamics. We provide enough details to enable the readers themselves to implement the presented methods without too much effort. 

    The idea behind operator splitting is certainly an old one and it has been comprehensively described in a number of publications. The new, and to some extent, original aspect of our presentation lies in the systematic use of numerical schemes and mathematical theory associated with hyperbolic problems. We use this hyperbolic approach to construct taylor-made operator-splitting methods, often allowing for large time- steps, and a unifying convergence theory that applies to discontinuous solutions. 

    The book is suitable for mathematicians, physicists, and engineers who wish to enlarge their knowledge about modern techniques using operator splitting to analyze and solve nonlinear partial differential equations of convection-diffusion type.