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 the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a "rough"coefficient function k(x) . We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, k' is in BV , thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general Lp compactness criterion.

/pdf/convergence-of-finite-difference-schemes-for-viscous-and-3lwzowmgzl.pdf

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

We study a model of continuous sedimentation. Under idealizing assumptions, the settling of the solid particles under the influence of gravity can be described by the initial value problem for a one-dimensional scalar conservation law with a flux function that depends discontinuously on the spatial position. We construct a weak solution to the sedimentation model by proving the convergence of a front tracking method. The basic building block in this method is the solution of the Riemann problem, which is complicated by the fact that the flux function is discontinuous. A feature of the convergence analysis is the difficulty of bounding the total variation of the conserved variable. To overcome this obstacle, we rely on a certain non-linear Temple functional under which the total variation can be bounded. The total variation bound on the transformed variable also implies that the front tracking construction is well defined. Finally, via some numerical examples, we demonstrate that the front tracking method can be used as a highly efficient and accurate simulation tool for continuous sedimentation.

A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units

We present a corrected operator splitting (COS) method for solving nonlinear parabolic equations of a convection-diffusion type. The main feature of this method is the ability to correctly resolve nonlinear shock fronts for large time steps, as opposed to a standard operator splitting (OS) which fails to do so. COS is based on solving a conservation law for modeling convection, a heat-type equation for modeling diffusion and finally a certain ``residual'' conservation law for necessary correction. The residual equation represents the entropy loss generated in the hyperbolic (convection) step. In OS the entropy loss manifests itself in the form of too wide shock fronts. The purpose of the correction step in COS is to counterbalance the entropy loss so that correct width of nonlinear shock fronts is ensured. The polygonal method of Dafermos [ J. Math. Anal. Appl., 38 (1972), pp. 33--41] constitutes an important part of our solution strategy. It is shown that COS generates a compact sequence of approximate solutions which converges to the solution of the problem. Finally, some numerical examples are presented where we compare OS and COS methods with respect to accuracy.

Corrected Operator Splitting for Nonlinear Parabolic Equations

We characterize the vanishing viscosity limit for multi-dimensional
conservation laws of the form
 
$ u_t + $div$ \mathfrak{f}(x,u)=0, \quad u|_{t=0}=u_0 $
 

in the domain $\mathbb R^+\times\mathbb R^N$. The flux $\mathfrak{f}=\mathfrak{f}(x,u)$ is assumed locally
Lipschitz continuous in the unknown $u$ and piecewise constant in the space variable $x$; the discontinuities of
$\mathfrak{f}(\cdot,u)$ are contained in the union of a locally finite number of sufficiently smooth
hypersurfaces of $\mathbb R^N$. We define "$\mathcal G_{VV}$-entropy solutions'' (this formulation is a
particular case of the one of [3]); the definition readily implies the uniqueness and the $L^1$
contraction principle for the $\mathcal G_{VV}$-entropy solutions. Our formulation is compatible with the
standard vanishing viscosity approximation
 
$ u^\varepsilon_t + $div$ (\mathfrak{f}(x,u^\varepsilon)) =\varepsilon \Delta u^\varepsilon, \quad
u^\varepsilon|_{t=0}=u_0, \quad \varepsilon\downarrow 0, $
 
of the conservation law. We show that, provided $u^\varepsilon$ enjoys an $\varepsilon$-uniform $L^\infty$ bound
and the flux $\mathfrak{f}(x,\cdot)$ is non-degenerately nonlinear, vanishing viscosity approximations
$u^\varepsilon$ converge as $\varepsilon \downarrow 0$ to the unique $\mathcal G_{VV}$-entropy solution of the
conservation law with discontinuous flux.

/pdf/on-vanishing-viscosity-approximation-of-conservation-laws-3p9g55uymo.pdf

On vanishing viscosity approximation of conservation laws with discontinuous flux

We study the Cauchy problem for the nonlinear (possibly strongly) degenerate parabolic transport-diffusion equation ∂tu+ ∂x ( γ(x)f(u) ) = ∂ xA(u), A ′(·) ≥ 0, where the coefficient γ(x) is possibly discontinuous and f(u) is genuinely nonlinear, but not necessarily convex or concave. Existence of a weak solution is proved by passing to the limit as e ↓ 0 in a suitable sequence {ue}e>0 of smooth approximations solving the problem above with the transport flux γ(x)f(·) replaced by γe(x)f(·) and the diffusion function A(·) replaced by Ae(·), where γe(·) is smooth and Ae(·) > 0. The main technical challenge is to deal with the fact that the total variation |ue|BV cannot be bounded uniformly in e, and hence one cannot derive directly strong convergence of {ue}e>0. In the purely hyperbolic case (A′ ≡ 0), where existence has already been established by a number of authors, all existence results to date have used a singular mapping to overcome the lack of a variation bound. Here we derive instead strong convergence via a series of a priori (energy) estimates that allow us to deduce convergence of the diffusion function and use the compensated compactness method to deal with the transport term.

/pdf/on-a-nonlinear-degenerate-parabolic-transport-diffusion-1tuzqoxpuu.pdf

Nils Henrik Risebro

Papers

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units

Corrected Operator Splitting for Nonlinear Parabolic Equations

On vanishing viscosity approximation of conservation laws with discontinuous flux

On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient