Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted into L loc ∞ estimates, following the Lip′ convergence theory developed by Tadmor et al. Comparisons between the local truncation error and the L loc ∞ -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41709/1/10444_2005_Article_7099.pd

Numerical methods for all-speed flows in fluid-dynamics and non-linear elasticity

In this thesis we are concerned with the numerical simulation of compressible materials flows, including gases, liquids and elastic solids. These materials are described by a monolithic Eulerian model of conservation laws, closed by an hyperelastic state law that includes the different behaviours of the considered materials. A novel implicit relaxation scheme to solve compressible flows at all speeds is proposed, with Mach numbers ranging from very small to the order of unity. The scheme is general and has the same formulation for all the considered materials, since a direct dependence on the state law is avoided via the relaxation. It is based on a fully implicit time discretization, easily implemented thanks to the linearity of the transport operator in the relaxation system. The spatial discretization is obtained by a combination of upwind and centered schemes in order to recover the correct numerical viscosity in different Mach regimes. The scheme is validated with one and two dimensional simulations of fluid flows and of deformations of compressible solids. We exploit the domain discretization through Cartesian grids, allowing for massively parallel computations (HPC) that drastically reduce the computational times on 2D test cases. Moreover, the scheme is adapted to the resolution on adaptive grids based on quadtrees, implementing adaptive mesh refinement techinques. The last part of the thesis is devoted to the numerical simulation of heterogeneous multi-material flows. A novel sharp interface method is proposed, with the derivation of implicit equilibrium conditions. The aim of the implicit framework is the solution of weakly compressible and low Mach flows, thus the proposed multi-material conditions are coupled with the implicit relaxation scheme that is solved in the bulk of the flow. Dans cette thèse on s’intéresse à la simulation numérique d’écoulements des matériaux compressibles, voir fluides et solides élastiques. Les matériaux considérés sont décrits avec un modèle monolithique eulérian, fermé avec une loi d’état hyperélastique qui considère les différents comportéments des matériaux. On propose un nouveau schéma de relaxation qui résout les écoulements compressibles dans des différents régimes, avec des nombres de Mach très petits jusqu’à l’ordre 1. Le schéma a une formulation générale qui est la même pour tous le matériaux considérés, parce que il ne dépend pas directement de la loi d’état. Il se base sur une discrétization complétement implicite, facile à implémenter grâce à la linearité de l’opérateur de transport du système de relaxation. La discrétization en éspace est donnée par la combinaison de flux upwind et centrés, pour retrouver la correcte viscosité numérique dans les différents régimes. L’utilisation de mailles cartésiennes pour les cas 2D s’adapte bien à une parallélisation massive, qui permet de réduire drastiquement le temps de calcul. De plus, le schéma a été adapté pour la résolution sur des mailles quadtree, pour implémenter l’adaptivité de la maille avec des critères entropiques. La dernière partie de la thèse concerne la simulation numérique d’écoulements multi-matériaux. On a proposé une nouvelle méthode d’interface “sharp”, en dérivant les conditions d’équilibre en implicite. L’objectif est la résolution d’interfaces physiques dans des régimes faiblement compressibles et avec un nombre de Mach faible, donc les conditions multi-matériaux sont couplées au schéma implicite de relaxation

Well-balanced computations of weak local residuals for the shallow water equations

The one-dimensional shallow water equations describe mass conservation and momentum conservation. We propose a well-balanced numerical technique for computing weak local residuals of the momentum equation. We compare the performance of weak local residuals of the momentum equation to those of the mass equation. All weak local residuals behave similarly. References E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein, and B. Perthame. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25:2050–2065, 2004. doi:10.1137/S1064827503431090. A. Bermudez and M. E. Vazquez. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids, 23:1049–1071, 1994. doi:10.1016/0045-7930(94)90004-3. L. A. Constantin and A. Kurganov. Adaptive central-upwind schemes for hyperbolic systems of conservation laws. In F. Asakura, S. Kawashima, A. Matsumura, S. Nishibata, K. Nishihara (Eds.) Hyperbolic problems: Theory, numerics, and applications. Vol. 1 of Proceedings of the 10th international conference, Osaka, Japan, 13–17 September 2004, Yokohama Publishers, Yokohama, 2006, pages 95–103, 2006. http://www.ybook.co.jp/pub/ISBN4-946552-21-9.htm, http://129.81.170.14/ kurganov/Constantin-Kurganov.pdf S. Karni and A. Kurganov. Local error analysis for approximate solutions of hyperbolic conservation laws. Adv. Comput. Math. 22:79–99, 2005. doi:10.1007/s10444-005-7099-8 S. Karni, A. Kurganov, and G. Petrova. A smoothness indicator for adaptive algorithms for hyperbolic systems. J. Comput. Phys. 178:323–341, 2002. doi:10.1006/jcph.2002.7024 R. Knobel. An Introduction to the Mathematical Theory of Waves. American Mathematical Society, Providence, 2000. http://www.ams.org/bookstore-getitem/item=stml-3 S. Mungkasi. A Study of Well-Balanced Finite Volume Methods and Refinement Indicators for the Shallow Water Equations. PhD thesis, Australian National University, Canberra, 2012. http://hdl.handle.net/1885/10301 S. Mungkasi and S. G. Roberts. On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface. ANZIAM J. 51:C48–C65, 2009 http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2576 S. Mungkasi and S. G. Roberts. Numerical entropy production for shallow water flows. ANZIAM J. 52:C1–C17, 2010. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3786 S. Noelle, N. Pankratz, G. Puppo, and J. R. Natvig. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phy. 213:474–499, 2006. doi:10.1016/j.jcp.2005.08.019 P. L. Roe. Characteristic-based schemes for the Euler equations. Ann. Rev. Fluid Mech. 18:337–365, 1986. doi:10.1146/annurev.fl.18.010186.002005 J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, New York, 1983. doi:10.1007/978-1-4612-0873-