Riemann Problems and Jupyter Solutions

by David I. Ketcheson, Randall J. LeVeque, and Mauricio J. del Razo

The Github repository containing these notebooks is https://github.com/clawpack/riemann_book.

You can view html versions of these notebooks at http://www.clawpack.org/riemann_book/html/Index.html.

Parts I and II of these notebooks will also be published by SIAM as a paperback book.

Please read about the License if you wish to use or modify this material.

This file and the rest of the files in the repository have been copied here for the convenience of ScienceData users, who can import individual notebooks or the whole directory to their own account and run the notebooks directly in your browser. Only this file and preface have been slightly altered to explain running the notebooks on ScienceData.

Contents

  • Preface -- Describes the aims and goals, and different ways to use the notebooks.

Part I: The Riemann problem and its solution

  1. Introduction -- Introduces basic ideas with some sample solutions.
  2. Advection -- The scalar advection equation is the simplest hyperbolic problem.
  3. Acoustics -- This linear system of two equations illustrates how eigenstructure is used.
  4. Burgers' equation -- The classic nonlinear scalar problem with a convex flux.
  5. Traffic flow -- A nonlinear scalar problem with a nice physical interpretation.
  6. Nonconvex_scalar -- More interesting Riemann solutions arise when the flux is not convex.
  7. Shallow water waves -- A classic nonlinear system of two equations.
  8. Shallow water with a tracer -- Adding a passively advected tracer and a linearly degenerate field.
  9. Euler equations of compressible gas dynamics -- The classic equations for an ideal gas.

Part II: Approximate solvers

  1. Approximate_solvers -- Introduction to two basic types of approximations.
  2. Burgers approximate -- Approximate solvers for a scalar problem.
  3. Shallow water_approximate -- Roe solvers, the entropy fix, positivity, HLL, and HLLE.
  4. Euler approximate -- Extension of these solvers to gas dynamics.
  5. Finite volume discretizations with approximate Riemann solvers -- Comparing how different approximate solvers perform when used with PyClaw.

Additional notebooks are under development and listed in Index2.ipynb. Updates may appear in the Github repository in the future.