**Description:**

Conservation
laws are the mathematical expression of the principles of conservation and
provide effective and accurate predictive models of our physical world.
Although intense research activity during the last decades has led to
substantial advances in the development of powerful computational methods for
conservation laws, their solution remains a challenge and many questions are
left open; thus it is an active and fruitful area of research.

*Numerical Methods for Conservation Laws: From Analysis to
Algorithms*

- offers
the first comprehensive introduction to modern computational methods and their
analysis for hyperbolic conservation laws, building on intense research
activities for more than four decades of development;
- discusses
classic results on monotone and finite difference/finite volume schemes, but
emphasizes the successful development of high-order accurate methods for
hyperbolic conservation laws;
- addresses
modern concepts of TVD and entropy stability, strongly stable Runge-Kutta
schemes, and limiter-based methods before discussing essentially nonoscillatory
schemes, discontinuous Galerkin methods, and spectral methods;
- explores
algorithmic aspects of these methods, emphasizing one- and two-dimensional problems
and the development and analysis of an extensive range of methods;
- includes
MATLAB software with which all main methods and computational results in the
book can be reproduced; and
- demonstrates
the performance of many methods on a set of benchmark problems to allow direct
comparisons.

Contents:

**Preface**

**MATLAB Scripts**

**Chapter 1:
Introduction** • Challenges ahead • Conservation through
history • Some of the great equations of continuum mechanics • Linear equations
• Nonlinear equations • Test cases and software • One-dimensional problems •
Two-dimensional problems • What remains and what needs to be found elsewhere •
Audience and use • References

__Part
I: Conservation Laws__

**Chapter 2: Scalar
conservation laws** • Weak solutions • Entropy conditions and
solutions • Entropy functions • References

**Chapter 3: Systems of
conservation laws** • Linear systems • Riemann problems •
Entropy conditions and functions • References

__Part
II: Monotone Schemes__

**Chapter 4: From
continuous to discrete** • Conservation and conservation form •
Monotonicity and entropy conditions • References

**Chapter 5: Finite
difference methods** • Linear problems • Consistency, stability,
and convergence • Nonsmooth problems • Nonlinear problems • Finite difference
methods in action • Boundary conditions • Linear wave equation • Burgers’
equation • Maxwell’s equations • Euler equations • References

**Chapter 6: Finite
volume methods** • Godunov’s method • Approximate Riemann
solvers • Roe fluxes • Engquist–Osher fluxes • Harten–Lax–van Leer (HLL) fluxes
• Central schemes • Finite volume methods in action • The Euler equations •
References

**Chapter 7: Beyond one
dimension** • Two-dimensional monotone schemes in action • Burgers’
equation • Nonconvex problem • The Euler equations • References

__Part
III: High-Order Schemes__

**Chapter 8: High-order
accuracy and its challenges** • The good • Phase error analysis • The
bad • Total variation stability • Entropy stability • The ugly • The Gibbs
phenomenon • Does it matter? • What to do if it does matter • References

**Chapter 9: Strong
stability preserving time integration** • Runge–Kutta methods •
Explicit strong stability preserving (SSP) Runge–Kuttaschemes • Implicit SSP
Runge–Kutta schemes • Order barriers • Multistep methods • References

**Chapter 10: High-order
accurate limiter-based methods **• Flux limited schemes •
Flux correction transport (FCT) schemes • TVD stable high-order schemes •
Positive schemes • Flux limited schemes in action • Slope limited schemes •
Monotone upstream-centered scheme for conservation laws (MUSCL) • Polynomial
methods based on Lagrangian reconstruction • Slope limited schemes in action •
Central schemes • Central schemes in action • Extension to multidimensional
problems • Burgers’ equation • Nonconvex scalar equation • Euler equations •
References

**Chapter 11:
Essentially nonoscillatory schemes** • Interpolation and
reconstruction • ENO methods • ENO method for conservation laws • A bit of
theory • ENO methods in action • WENO methods • WENO variants • Well balanced
schemes • A little more theory • WENO methods in action • Dealing with
nonuniform grids • Beyond one dimension • Interlude on non-Cartesian boundaries
• Scalar equations • The Euler equations • References

**Chapter 12:
Discontinuous Galerkin methods** • The basics • The local approximation
• Key properties • Error estimates • Phase error analysis • Nonsmooth problems
• A detour on hidden accuracy • Filtering • Nonlinear dissipation • Slope
limiters • WENO-based limiters • Extrema preserving limiters • Related
formulations • Spectral penalty methods • Spectral finite volume schemes •
Spectral difference schemes • Flux reconstruction schemes • Extension to
multidimensional problems • Discontinuous Galerkin methods in action • Linear
wave equation • Burgers’ equation • Maxwell’s equations • The Euler equations •
References

**Chapter 13: Spectral
methods** • Fourier modes and nodes • The continuous Fourier
expansion • The discrete Fourier expansion • Fourier spectral methods •
Fourier–Galerkin methods • Fourier collocation methods • Nonlinear problems •
Skew-symmetric form • Vanishing viscosity • Postprocessing • Filtering •
Fourier–Padé reconstruction • Overcoming the Gibbs phenomenon • Spectral
methods in action • Burgers’ equation • Maxwell’s equations • Euler equations

• References

**Index**

About the Author:

**Jan S. Hesthaven** is Dean of Basic Sciences,
Professor of Mathematics, and holds the Chair of Computational Mathematics and
Simulation Science at Ecole Polytechnique Fédérale de Lausanne (EPFL) in
Switzerland. Prior to joining EPFL in 2013, he was Professor of Applied
Mathematics at Brown University. He has worked for more than two decades on the
development, analysis, and application of modern computational methods for
linear and nonlinear wave problems, with an emphasis on high-order accurate
methods. He is an Alfred P. Sloan Fellow (2001), an NSF Career award winner
(2002), and a SIAM Fellow (2014).

Target Audience:

This
book is intended for graduate students in computational mathematics and
researchers seeking a comprehensive introduction to modern methods for solving
conservation laws. Students and researchers in applied sciences and engineering
will benefit from the book’s emphasis on algorithmic aspects of complex
algorithms. The text also includes extensive references which allows
researchers to pursue advanced research and results.