Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

https://depts.washington.edu/clawpack/book2/sample.pdf

Finite Volume Methods for Hyperbolic Problems

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations

Stochastic equations in infinite dimensions

In this edition we have added the following new material: In Chapt. 1 we have added
a section on linear equations, which allows us to present some of the material in the
book in the simpler linear setting. In Chapt. 2 we have made some changes in the
presentation of Kružkov’s fundamental doubling of variables method. In Chapt. 3
on finite difference methods the focus has been changed to finite volume methods.
A section on higher-order schemes has been added. The section on measure-valued
solutions has been rewritten. The main existence theorem in Chapt. 4, Theorem 4.3,
now resembles the one-dimensional case. The presentation of the solution of the
Riemann problem for systems in Chapt. 5 has been supplemented by the complete
solution of the Riemann problem for the 3 3 Euler equations of gas dynamics.
The solution of the Cauchy problem for systems in Chapt. 6 has been rewritten
and simplified. We have added a new chapter, Chapt. 8, on one-dimensional scalar
conservation laws where the flux function depends explicitly on space in a discontinuous manner

Front Tracking for Hyperbolic Conservation Laws

We introduce a model that describes heavy traffic on a network of unidirectional roads. The model consists of a system of initial-boundary value problems for nonlinear conservation laws. We uniquely formulate and solve the Riemann problem for such a system and, based on this, then show existence of a solution to the Cauchy problem.

/pdf/a-mathematical-model-of-traffic-flow-on-a-network-of-2dl85qhhd6.pdf

A mathematical model of traffic flow on a network of unidirectional roads

Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs

We study nonlinear degenerate parabolic equations where the
 flux function $f(x,t,u)$ does not depend Lipschitz
 continuously on the spatial location $x$.
 By properly adapting the "doubling of variables" device due
 to Kružkov [25] and Carrillo [12], we prove
 a uniqueness result within the class of
 entropy solutions for the initial value problem.
 We also prove a result concerning the continuous dependence
 on the initial data and the flux function for degenerate
 parabolic equations with flux function of the form
 $k(x)f(u)$, where $k(x)$ is a vector-valued function
 and $f(u)$ is a scalar function.

/pdf/on-the-uniqueness-and-stability-of-entropy-solutions-of-35fy3dxud8.pdf

On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

We propose a general framework for the study of L1 contractive semigroups of solutions to conservation laws with discontinuous flux: 

$$ u_t + \mathfrak{f}(x,u)_x=0, \qquad \mathfrak{f}(x,u)= \left\{\begin{array}{ll} f^l(u),& x 0, \end{array} \right\quad\quad\quad (\rm CL) $$

where the fluxes fl, fr are mainly assumed to be continuous Developing the ideas of a number of preceding works (Baiti and Jenssen in J Differ Equ 140(1):161–185, 1997; Towers in SIAM J Numer Anal 38(2):681–698, 2000; Towers in SIAM J Numer Anal 39(4):1197–1218, 2001; Towers et al in Skr K Nor Vidensk Selsk 3:1–49, 2003; Adimurthi et al in J Math Kyoto University 43(1):27–70, 2003; Adimurthi et al in J Hyperbolic Differ Equ 2(4):783–837, 2005; Audusse and Perthame in Proc Roy Soc Edinburgh A 135(2):253–265, 2005; Garavello et al in Netw Heterog Media 2:159–179, 2007; Burger et al in SIAM J Numer Anal 47:1684–1712, 2009), we claim that the whole admissibility issue is reduced to the selection of a family of “elementary solutions”, which are piecewise constant weak solutions of the form 

$$ c(x)=c^l11_{\left\{{x 0}\right\}} $$

We refer to such a family as a “germ” It is well known that (CL) admits many different L1 contractive semigroups, some of which reflect different physical applications We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions We devote specific attention to the “vanishing viscosity” germ, which is a way of expressing the “Γ-condition” of Diehl (J Hyperbolic Differ Equ 6(1):127–159, 2009) For any given germ, we formulate “germ-based” admissibility conditions in the form of a trace condition on the flux discontinuity line {x = 0} [in the spirit of Vol’pert (Math USSR Sbornik 2(2):225–267, 1967)] and in the form of a family of global entropy inequalities [following Kruzhkov (Math USSR Sbornik 10(2):217–243, 1970) and Carrillo (Arch Ration Mech Anal 147(4):269–361, 1999)] We characterize those germs that lead to the L1-contraction property for the associated admissible solutions Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities “adapted” to the choice of a germ), or for specific germ-adapted finite volume schemes

/pdf/a-theory-of-l-1-dissipative-solvers-for-scalar-conservation-1i23lfwa38.pdf

Nils Henrik Risebro

Papers

Front Tracking for Hyperbolic Conservation Laws

A mathematical model of traffic flow on a network of unidirectional roads

Splitting methods for partial differential equations with rough solutions : analysis and MATLAB programs

On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients

A Theory of L 1 -Dissipative Solvers for Scalar Conservation Laws with Discontinuous Flux