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

Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models

We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples. AMS subject classifications: 76M12, 65B99, 35L65

Quasi-Lagrangian Acceleration of Eulerian Methods

We present an adaptive well-balanced positivity preserving scheme on quadtree grids for shallow water equations. The use of quadtree grids results in a robust, efficient and highly accurate numerical method. The quadtree model is developed based on the well-balanced positivity preserving central-upwind scheme proposed in [{\sc A. Kurganov and G. Petrova}, {\em Commun. Math. Sci.}, 5 (2007), pp. 133--160]. The designed scheme is well-balanced in the sense that it is capable of exactly preserving "lake-at-rest" steady states. In order to achieve this as well as to preserve positivity of water depth, a continuous piecewise bi-linear interpolation of the bottom topography function is utilized. Local gradients are examined to determine new seeding points in grid refinement for the next timestep. Numerical examples demonstrate the promising performance of the central-upwind quadtree scheme.

An adaptive well-balanced positivity preserving scheme on quadtree grids for shallow water equations.

https://link.springer.com/content/pdf/10.1007/s42967-021-00161-2.pdf

Alexander Kurganov

Papers

Fifth-Order A-WENO Path-Conservative Central-Upwind Scheme for Behavioral Non-Equilibrium Traffic Models

Quasi-Lagrangian Acceleration of Eulerian Methods

An adaptive well-balanced positivity preserving scheme on quadtree grids for shallow water equations.

Fifth-Order A-WENO Schemes Based on the Adaptive Diffusion Central-Upwind Rankine-Hugoniot Fluxes

A new sticky particle method for pressureless gas dynamics