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

https://hal.archives-ouvertes.fr/hal-01634261/document

Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy

In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the Hamilton--Jacobi equation: $$ \phi_t + H(x_1,\ldots,x_d,t,\phi,\phi_{x_1},\ldots,\phi_{x_d}) = 0. $$ This weighted ENO scheme is constructed upon and has the same stencil nodes as the third order ENO scheme but can be as high as fifth order accurate in the smooth part of the solution. In addition to the accuracy improvement, numerical comparisons between the two schemes also demonstrate that the weighted ENO scheme is more robust than the ENO scheme.

Weighted ENO Schemes for Hamilton--Jacobi Equations

After several years of planning, the 1st International Workshop on High-Order CFD Methods was successfully held in Nashville, Tennessee, on January 7-8, 2012, just before the 50th Aerospace Sciences Meeting. The American Institute of Aeronautics and Astronautics, the Air Force Office of Scientific Research, and the German Aerospace Center provided much needed support, financial and moral. Over 70 participants from all over the world across the research spectrum of academia, government labs, and private industry attended the workshop. Many exciting results were presented. In this review article, the main motivation and major findings from the workshop are described. Pacing items requiring further effort are presented. © 2013 John Wiley & Sons, Ltd.

/pdf/high-order-cfd-methods-current-status-and-perspective-1hm5p7dcff.pdf

High-order CFD methods: Current status and perspective

Weighted essentially non-oscillatory schemes

This is the first paper in a series in which a class of nonoscillatory high order accurate self-similar local maximum principle satisfying (in scalar conservation law) shock capturing schemes for solving multidimensional systems of conservation laws are constructed and analyzed. In this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented, using scalar one-dimensional initial value problems as a model. For this model, the schemes are made to satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modifications, imposed as needed. The first enforces a local maximum principle; the second guarantees that no new extrema develop. The schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [Math. Comp., 49 (1987), pp. 105–121, Math. Comp...

Xu-Dong Liu

Papers

Weighted essentially non-oscillatory schemes

Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I

Giant gastrointestinal stromal tumor of the jejunum combine with spontaneous rupture and hemorrhage: A case report.