Preface to the special issue "Recent progress in numerical methods for nonlinear time-dependent flow & transport problems"

This special issue is dedicated to recent developments concerning modern numerical methods for time dependent partial differential equations that can be used to simulate complex nonlinear flow and transport processes and tries to cover a wide spectrum of different applications and numerical approaches. Most of the models under consideration here are nonlinear hyperbolic conservation laws, which are a very powerful mathematical tool to describe a large set of problems in science and engineering. Hyperbolic PDE are based on first principles, such as mass, momentum and total energy conservation as well as the second law of thermodynamics. These principles are universally valid and have been successfully used for the simulation of a very wide class of problems, ranging from astrophysics over geophysics to complex flows in engineering and computational biology. Most of the seemingly different applications listed above have a common mathematical description under the form of nonlinear hyperbolic systems of partial differential equations, possibly containing also higher-order derivative terms, non-conservative products and nonlinear (potentially stiff) source terms. From the mathematical point of view, the major difficulties in these systems arise from the nonlinearities that allow the formation of non-smooth solutions such as shock waves, material interfaces or other discontinuities, but nonlinear conservation laws can simultaneously also contain smooth features like sound waves and vortex flows. Even nowadays the key challenge for the construction of successful numerical methods for nonlinear hyperbolic PDE is therefore to capture at the same time shock waves and discontinuities robustly without producing spurious oscillations and exhibit little numerical dissipation and dispersion errors in smooth regions of the solution. For this reason, the development of robust and accurate numerical methods for nonlinear flow and transport problems remains a great challenge even after several decades of successful research, which started back in the 1940s and 1950s with the pioneering works of von Neumann [1] and Godunov [2]. The present special issue is published on the occasion of the 70th birthday of Prof. Eleuterio F. Toro and in honor of his outstanding contributions to the fields of applied mathematics, scientific computing and computational physics with his well-known numerical methods for the solution of hyperbolic conservation laws and their application to computational fluid dynamics, computational physics and computational biology. (C) 2018 Published by Elsevier Ltd.