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Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB® Programs
Helge Holden, Norwegian University of Science and Technology, Trondheim, Norway, and Kenneth H. Karlsen, Knut-Andreas Lie, and Nils Henrik Risebro, University of Oslo, Norway
A publication of the European Mathematical Society.
cover
EMS Series of Lectures in Mathematics
2010; 236 pp; softcover
Volume: 11
ISBN-10: 3-03719-078-7
ISBN-13: 978-3-03719-078-4
List Price: US$48
Member Price: US$38.40
Order Code: EMSSERLEC/11
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Operator splitting (or the fractional steps method) is a very common tool to analyze nonlinear partial differential equations both numerically and analytically. By applying operator splitting to a complicated model one can often split it into simpler problems that can be analyzed separately. In this book one studies operator splitting for a family of nonlinear evolution equations, including hyperbolic conservation laws and degenerate convection-diffusion equations. Common for these equations is the prevalence of rough, or non-smooth, solutions, e.g., shocks.

Rigorous analysis is presented, showing that both semi-discrete and fully discrete splitting methods converge. For conservation laws, sharp error estimates are provided and for convection-diffusion equations one discusses a priori and a posteriori correction of entropy errors introduced by the splitting. Numerical methods include finite difference and finite volume methods as well as front tracking. The theory is illustrated by numerous examples. There is a dedicated Web page that provides MATLAB® codes for many of the examples.

The book is suitable for graduate students and researchers in pure and applied mathematics, physics, and engineering.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

® MATLAB, The MathWorks, Inc., Natick, MA.

Readership

Graduate students and research mathematicians interested in partial differential equations.

Table of Contents

  • Introduction
  • Simple examples of semi-discrete operator splitting
  • General convergence theory
  • Convergence results for convection-diffusion problems
  • Error estimates for hyperbolic problems
  • Operator splitting for systems of equations
  • A. A crash course in numerical methods for conservation laws
  • References
  • Index
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