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

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

High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

https://ntrs.nasa.gov/api/citations/19980201047/downloads/19980201047.pdf

Low-Dissipative High-Order Shock-Capturing Methods Using Characteristic-Based Filters

Studies in Advanced Mathematics

Capturing Shock Reflections

Data‐dependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed “a la Harten”. In this paper we carefully analyze the class of Essentially Non‐Oscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.

Nonlinear multiscale decompositions: The approach of A. Harten

A Flux-Split Algorithm Applied to Relativistic Flows

A nonlinear multiresolution scheme within Harten's framework is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

We perform a computational study of the interaction of a planar shock wave with a cylindrical vortex. We use a particularly robust High Resolution Shock Capturing scheme, Marquina's scheme, to obtain high quality, high resolution numerical simulations of the interaction. In the case of a very-strong shock/vortex encounter, we observe a severe reorganization of the flow field in the downstream region, which seems to be due mainly to the strength of the shock. The numerical data is analyzed to study the driving mechanisms for the production of vorticity in the interaction.

/pdf/shock-vortex-interactions-at-high-mach-numbers-3evrthc6ml.pdf

Rosa Donat

Papers

Capturing Shock Reflections

Nonlinear multiscale decompositions: The approach of A. Harten

A Flux-Split Algorithm Applied to Relativistic Flows

Analysis of a New Nonlinear Subdivision Scheme. Applications in Image Processing

Shock-Vortex Interactions at High Mach Numbers