Title Numerical Methods for Conservation Laws
Subtitle From Analysis to Algorithm
Author Jan S. Hesthaven
ISBN 9781611975093
List price USD 89.00
Price outside India Available on Request
Original price
Binding Paperback
No of pages 576
Book size 153 x 229 mm
Publishing year 2018
Original publisher SIAM - Society for Industrial and Applied Mathematics (Eurospan Group)
Published in India by .
Exclusive distributors Viva Books Private Limited
Sales territory India, Sri Lanka, Bangladesh, Pakistan, Nepal, .
Status New Arrival
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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.



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


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.


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