Riemann Problems and Jupyter Solutions (Fundamentals of Algorithms)

This book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity. The solution of the Riemann problem captures essential information about these models and is the key ingredient in modern numerical methods for their solution. This book covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem. The emphasis is on the general ideas, but each chapter delves into a particular application. Riemann Problems and Jupyter Solutions is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations, allowing readers to grasp how the concepts presented are affected by important parameters and to experiment by varying those parameters themselves. The only interactive book focused entirely on the Riemann problem, it develops each concept in the context of a specific physical application, helping readers apply physical intuition in learning mathematical concepts. Graduate students and researchers working in the analysis and/or numerical solution of hyperbolic PDEs will find this book of interest. This includes mathematicians, as well as scientists and engineers, working on wave propagation problems. Educators interested in developing instructional materials using Jupyter notebooks will also find this book useful. The book is appropriate for courses in Numerical Methods for Hyperbolic PDEs and Analysis of Hyperbolic PDEs, and it can be a great supplement for courses in computational fluid dynamics, acoustics, and gas dynamics.

David I. Ketcheson is an associate professor of applied mathematics and computational science at King Abdullah University of Science & Technology and a lead developer of the Clawpack and PyClaw software projects. His research interests include numerical analysis, scientific software, and analysis of physical wave phenomena. Randall J. LeVeque is Professor Emeritus of Applied Mathematics at the University of Washington. He is a lead developer of the Clawpack and GeoClaw software packages and the author of several books on numerical methods for differential equations. His current research interests focus on software development and hazard assessment, particularly for tsunamis and other geophysical flows. He is a Fellow of SIAM and the American Mathematical Society. Mauricio J. del Razo is a postdoctoral researcher at the Freie Universität Berlin. His research interests range from numerical methods for conservation laws to multiscale reaction-diffusion processes. He is also interested in using and developing new technologies for teaching and communicating science.