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

We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

/pdf/viscosity-solutions-of-hamilton-jacobi-equations-with-1fsg24i1sm.pdf

Viscosity solutions of hamilton–jacobi equations with discontinuous coefficients

The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem 

$$\displaystyle u_{t}+f(u)_{x}=0,\quad u(x,0)=u_{0}(x),$$

(7.1)

is stable in L 1 with respect to perturbations in the initial data. In other words, if $v=v(x,t)$ is another solution found by front tracking, then 

$$\displaystyle{\left\|u(\,\cdot\,,t)-v(\,\cdot\,,t)\right\|}_{1}\leq C{\left\|u_{0}-v_{0}\right\|}_{1}$$

for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.

Well-Posedness of the Cauchy Problem

Clarifier-thickener units treating ideal suspensions can be modeled as an initial-value problem for a nonconvex scalar conservation law whose flux depends on a vector of discontinuous parameters. This problem can be treated by the well-known Engquist–Osher scheme if the discontinuous paremeters are discretized on a grid staggered against that of the conserved variable. We prove convergence of this scheme to a weak solution of the problem and illustrate its application to the clarifier-thickener setup by a numerical example.

Monotone difference approximations for the simulation of clarifier-thickener units

We offer a simple and self-contained proof that the Follow-the-Leader model converges to the Lighthill-Whitham-Richards model for traffic flow.

Continuum limit of Follow-the-Leader models – a short proof

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. ...

/pdf/a-multilevel-monte-carlo-finite-difference-method-for-random-2unaezvi7k.pdf

Nils Henrik Risebro

Papers

Viscosity solutions of hamilton–jacobi equations with discontinuous coefficients

Well-Posedness of the Cauchy Problem

Monotone difference approximations for the simulation of clarifier-thickener units

Continuum limit of Follow-the-Leader models – a short proof

A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations