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).

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

We study a class of degenerate parabolic convection–diffusion equations, endowed with a mechanism for saturation of the diffusion flux, which corrects the unphysical gradient-flux relations at high gradients. This paper extends our previous works on the effects of diffusion with saturation on convection and the impact of saturation on porous media-type diffusion, where it has been demonstrated that a nonlinear saturating diffusion is susceptible to a self-induced formation of discontinuities. In this work we demonstrate that nonlinear convection enhances the breakdown effect. We carry both analytical and numerical studies of the model equation, ut + f(u)x = [(u)Q(ux, u)]x, where Q is a bounded increasing function, (0) = 0 and (u) ~ un, n > 0 for u ~ 0. Depending on a choice of n, we obtain two distinctive processes. If 0 ≤ n ≤ 1, a discontinuity forms only when the upstream–downstream disparity exceeds a critical threshold, but if n > 1, all travelling waves are found to have a sharp discontinuous front. In fact, given a compact or a semi-compact initial datum, the front will not start to move until such a discontinuity forms.

https://iopscience.iop.org/article/10.1088/0951-7715/18/2/009/pdf

On degenerate saturated-diffusion equations with convection

A new adaptive diffusion central numerical flux within the framework of fifth-order characteristicwise alternative WENO-Z finite-difference schemes (A-WENO) with a modified local Lax--Friedrichs (L...

Fifth-Order A-WENO Finite-Difference Schemes Based on a New Adaptive Diffusion Central Numerical Flux

The trapezoidal quadrature rule on a uniform grid has spectral accuracy when integrating C ∞ periodic function over a period. The same holds for quadrature formulae based on piecewise polynomial interpolations. In this paper, we prove that these quadratures applied to \({\rm{W}}_{{\rm{per}}}^{{\rm{r,p}}} \) periodic functions with r > 2 and p ≥ 1 have error \({\mathcal O}((\Delta x)^r)\). The order is independent of p, sharp, and for p < ∞ is higher than predicted by best trigonometric approximation. For p=1 it is higher by 1.

/pdf/the-order-of-accuracy-of-quadrature-formulae-for-periodic-7wnmiikwkh.pdf

The Order of Accuracy of Quadrature Formulae for Periodic Functions

In this paper, we describe how to construct a finite-difference shock-capturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domain Ω, possibly with moving boundary. The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls. Although the motion of the boundary could be coupled with the fluid, all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow. The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domain ΩR ⊃ Ω. We identify inner and ghost points. The inner points are the grid points located inside Ω, while the ghost points are the grid points that are outside Ω but have at least one neighbor inside Ω. The evolution equations for inner points data are obtained from the discretization of the governing equation, while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables (density, velocities and pressure). Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains. Several numerical experiments are conducted to illustrate the validity of the method. We demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

/pdf/a-second-order-finite-difference-method-for-compressible-2u42m203c9.pdf

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

We review a class of Godunov-type finite-volume methods for hyperbolic systems of conservation and balance laws—nonoscillatory central schemes. These schemes date back to 1950s, when the first-order Lax–Friedrichs scheme was introduced. The central Lax–Friedrichs scheme can be viewed as a simple alternative to the upwind Godunov scheme, which was also introduced in the 1950s. The main idea in the construction of both central and upwind first-order schemes is the same: use a global piecewise constant approximation of the solution at a certain time level and evolve it in time to the next time level exactly. The exact evolution is performed using the integral form of the studied system of PDEs. The difference is in one small detail—which is in fact not small at all—how to select the space–time control volume for the time evolution. The key idea in the construction of central schemes is to choose these control volume in such a way that no (localized) Riemann problems need to be solved at the evolution step. This makes central schemes particularly simple and universal numerical tool for general hyperbolic systems. On the other hand, central schemes are based on averaging the nonlinear waves rather than resolving them and thus they have larger numerical dissipation than their upwind counterparts. In order to increase the resolution achieved by central schemes, one has to increase their order. We describe how to design high-order nonoscillatory central schemes and also discuss how to further decrease their numerical dissipation without risking oscillations. The latter is achieved by utilizing some upwinding information (local speeds of propagation) within the framework of the Riemann-problem-solver-free central schemes and modifying the set of control volumes used for the time evolution. This leads to another type of central schemes—central-upwind schemes, whose derivation is reviewed in this work.

Alexander Kurganov

Papers

On degenerate saturated-diffusion equations with convection

Fifth-Order A-WENO Finite-Difference Schemes Based on a New Adaptive Diffusion Central Numerical Flux

The Order of Accuracy of Quadrature Formulae for Periodic Functions

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

Central Schemes: A Powerful Black-Box Solver for Nonlinear Hyperbolic PDEs