Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori-Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel , using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto-Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.
In this work, the space of admissible entropy functions for the compressible multicomponent Euler equations is explored, following up on [Harten, J. Comput. Phys., 49 (1), 1983, pp. 151-164]. This effort allows us to prove a minimum entropy principle on entropy solutions, whether smooth or discrete, in the same way it was originally demonstrated for the compressible Euler equations by [Tadmor, Appl. Numer. Math., 49 (3-5), 1986, pp. 211-219].
In this work, Entropy-Stable (ES) schemes are formulated for the multicomponent compressible Euler equations. Entropy-conservative (EC) and ES fluxes are derived. Particular attention is paid to the limit case of zero partial densities where the structure required by ES schemes breaks down (the entropy variables are no longer defined). It is shown that while an EC flux is well-defined in this limit, a well-defined upwind ES flux requires appropriately averaged partial densities in the dissipation matrix. A similar result holds for the high-order TecNO reconstruction. However, this does not prevent the numerical solution from developing negative partial densities or internal energy. Numerical experiments were performed on one-dimensional and two-dimensional interface and shock-interface problems. The present scheme exactly preserves stationary interfaces. On moving interfaces, it produces spurious pressure oscillations typically observed with conservative schemes [Karni, J. Comput. Phys., 112 (1994) 1]. We find that these anomalies, which are not present in the single-component case, violate neither entropy stability nor a minimum principle of the specific entropy. Finally, we show that the scheme is able to reproduce the physical mechanisms of the two-dimensional shock-bubble interaction problem [Haas & Sturtevant,
This work delves into the family of entropy conservative (EC) schemes introduced by Tadmor. The discussion is centered around the Euler equations of fluid mechanics and the receding flow problem extensively studied by Liou. This work is motivated by Liou's recent findings that an abnormal spike in temperature observed with finite-volume schemes is linked to a spurious entropy rise, and that it can be prevented in principle by conserving entropy. While a semi-discrete analysis suggests EC schemes are a good fit, a fully discrete analysis based on Tadmor's framework shows the non-negligible impact of time-integration on the solution behavior. An EC time-integration scheme is developed to show that enforcing conservation of entropy at the fully discrete level does not necessarily guarantee a well-behaved solution.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.