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 numerical methods for convection-dominated fluid dynamics problems. In particular, we consider initial-boundary value problems for the Burgers equation with small diffusion coefficients. Our goal is to investigate several strategies, which can be used to monotonize numerical methods and to ensure non-oscillatory and positivity-preserving properties of the computed solutions. We focus on fully implicit finite-element methods constructed using the backward Euler time discretization combined with high-order spatial approximations. We experimentally study the following three monotonization approaches: mesh refinement, increasing the time-step size and utilizing higher-order finite-element approximations. Feasibility of these three strategies is demonstrated on a number of numerical examples for both one- and two-dimensional Burgers equations.

Monotonization of a Family of Implicit Schemes for the Burgers Equation

We develop new hybrid numerical algorithms for compressible multicomponent fluids problem. The fluid components are assumed to be immiscible and are separated by material interface. We track the location of the interface using the level set approach and replace the energy equation in the original model with the corresponding pressure equation in its neighborhoods. In these neighboring areas we solve the resulting nonconservative system using a path-conservative central-upwind scheme, while in the rest of the computational domain, a central-upwind scheme is used to numerically solve the original conservative system. We first develop a finite-volume method of the second order and then extend it to the fifth order via the finite-difference alternative WENO (A-WENO) framework. In order to reduce oscillations, we switch from A-WENO back to second-order central-upwind scheme in certain nonsmooth parts of the computational solution. We illustrate the performance of the new hybrid methods on a number of one- and two-dimensional examples including the shock–bubble interaction tests.

Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme

We develop a new second-order unstaggered path-conservative central-upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, it does not rely on any (approximate) Riemann problem solver, and it robustly produces high-resolution and non-oscillatory results. The derivation of the scheme is based on the Godunov-Powell nonconservative modiﬁcations of the studied MHD systems. The local divergence-free property is enforced by augmenting the modiﬁed systems with the evolution equations for the corresponding derivatives of the magnetic ﬁeld components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic ﬁeld, which guarantees a non-oscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. We test the proposed PCCU scheme on several benchmarks for both ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution, but also preserve the positivity of computed quantities such as density, pressure, and water depth.

A New Locally Divergence-Free Path-Conservative Central-Upwind Scheme for Ideal and Shallow Water Magnetohydrodynamics

We develop new adaptive alternative weighted essentially non-oscillatory (A-WENO) schemes for hyperbolic systems of conservation laws. The new schemes employ the recently proposed local characteristic decomposition based central-upwind numerical fluxes, the three-stage third-order strong stability preserving Runge-Kutta time integrator, and the fifth-order WENO-Z interpolation. The adaptive strategy is implemented by applying the limited interpolation only in the parts of the computational domain where the solution is identified as ``rough'' with the help of a smoothness indicator. We develop and use a new simple and robust local smoothness indicator (LSI), which is applied to the solutions computed at each of the three stages of the ODE solver. The new LSI and adaptive A-WENO schemes are tested on the Euler equations of gas dynamics. We implement the proposed LSI using the pressure, which remains smooth at contact discontinuities, while our goal is to detect other ``rough'' areas and apply the limited interpolation mostly in the neighborhoods of the shock waves. We demonstrate that the new adaptive schemes are highly accurate, non-oscillatory, and robust. They outperform their fully limited counterparts (the A-WENO schemes with the same numerical fluxes and ODE solver but with the WENO-Z interpolation employed everywhere) while being less computationally expensive.

Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator

We consider a two-dimensional pedestrian flow model with obstacles governed by scalar hyperbolic conservation laws, in which the flux is implicitly dependent on the density through the Eikonal equation. We propose a simple second-order finite-volume method, which is applicable to the case of obstacles of arbitrary shapes. Though the method is only first-order accurate near the obstacles, it is robust and provides sharp resolution of discontinuities as illustrated in a number of numerical experiments.

Alexander Kurganov

Papers

Monotonization of a Family of Implicit Schemes for the Burgers Equation

Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme

A New Locally Divergence-Free Path-Conservative Central-Upwind Scheme for Ideal and Shallow Water Magnetohydrodynamics

Adaptive High-Order A-WENO Schemes Based on a New Local Smoothness Indicator

A Simple Finite-Volume Method on a Cartesian Mesh for Pedestrian Flows with Obstacles