Numerical Schemes for Conservation Laws

Many problems of fluid mechanics, in particular in gas-dynamics, have led to the study of non-linear hyperbolic conservation laws. Since the solutions of conservation laws may be discontinuous, even for smooth data, special stabilization techniques such as upwinding are necessary. This book systematically studies upwind methods for initial value problems for scalar conservation laws in one- and multidimensions. The mathematical theory of convergence theory and of a priori error estimates is presented in detail for structured grids (finite difference methods) as well as for unstructured grids (finite volume methods). Higher order schemes are also included. In the second part of this book the algorithms for scalar equations are generalized into systems of conservation laws in one- and multidimensions. The most powerful schemes for the discretization of systems are described and numerical examples are presented. In particular, local grid refinement has been taken into account.The initial boundary value problem is also considered for linear systems and nonlinear scalar conservation laws. Since nearly all mathematical details are developed and described in this book, it can be used as an introduction into the theory of upwind methods for conservation laws.