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

Boundary conditions for direct simulations of compressible viscous flows

This comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book, which can also form the basis of one or more graduate-level courses.

/pdf/atmospheric-modeling-data-assimilation-and-predictability-1qxqadvqdq.pdf

Atmospheric Modeling, Data Assimilation and Predictability

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

https://www3.nd.edu/~zxu2/acms60790S13/Shu-WENO-notes.pdf

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

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

Praise for the First Edition

	". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations."
	—SIAM Review

	Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods.

	The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes:

	
		High order methods on staggered grids
	
		Extended treatment of Summation By Parts operators and their application to second-order derivatives
	
		Simplified presentation of certain parts and proofs


	Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.

Time Dependent Problems and Difference Methods

Preface to the Classics Edition Introduction 1. The Navier-Stokes equations 2. Constant-coefficient Cauchy problems 3. Linear variable-coefficient Cauchy problems in 1D 4. A nonlinear example: Burgers' equations 5. Nonlinear systems in one space dimension 6. The Cauchy problem for systems in several dimensions 7. Initial-boundary value problems in one space dimension 8. Initial-boundary value problems in several space dimensions 9. The incompressible Navier-Stokes equations: the spatially periodic case 10. The incompressible Navier-Stokes equations under initial and boundary conditions Appendices References Author index Subject index.

Initial-boundary value problems and the Navier-Stokes equations

It is shown that the dilatational terms that need to be modeled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of these dilatational terms in homogeneous turbulence is explored by asymptotic analysis of the compressible Navier-Stokes equations. A non-dimensional parameter which characterizes some compressible effects in moderate Mach number, homogeneous turbulence is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer.

https://ntrs.nasa.gov/api/citations/19900005197/downloads/19900005197.pdf

The analysis and modelling of dilatational terms in compressible turbulence

Stability theory of difference approximations for mixed initial boundary value problems. II

/pdf/stability-theory-of-difference-approximations-for-mixed-4qsfcrjuir.pdf

Compressible turbulent flows at low turbulent Mach numbers are considered Contrary to the general belief that such flows are almost incompressible, (ie, the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared These results are verified by extensive direct numerical simulations of isotropic turbulence

Heinz-Otto Kreiss

Papers

Time Dependent Problems and Difference Methods

Initial-boundary value problems and the Navier-Stokes equations

The analysis and modelling of dilatational terms in compressible turbulence

Stability theory of difference approximations for mixed initial boundary value problems. II

The analysis and simulation of compressible turbulence