## HLLE

 swMATH ID: 6605 Software Authors: Harten, Amiram; Lax, Peter D.; van Leer, Bram Description: The HLLE[3] (Harten, Lax, van Leer and Einfeldt) solver is an approximate solution to the Riemann problem, which is only based on the integral form of the conservation laws and the largest and smallest signal velocities at the interface. The stability and robustness of the HLLE solver is closely related to the signal velocities and a single central average state, as proposed by Einfeldt in the original paper. The description of the HLLE scheme in the book mentioned below is incomplete and partially wrong. The reader is referred to the original paper. Actually, the HLLE scheme is based on a new stability theory for discontinuities in fluids, which was never published. HLLC solver The HLLC (Harten-Lax-van Leer-Contact) solver was introduced by Toro.[4] It restores the missing Rarefaction wave by some estimates, like linearisations, these can be simple but also more advanced exists like using the Roe average velocity for the middle wave speed. They are quite robust and efficient but somewhat more diffusive.[5]https://math.nyu.edu/ jbu200/E1GODF.F Homepage: http://en.wikipedia.org/wiki/Riemann_solver#HLLE_solver Keywords: Riemann solver Related Software: HE-E1GODF; AUSM; RIEMANN; HLLC; MOOD; AUSMPW+; SHASTA; PVM; Athena; FLASH; SWASHES; Pluto; ECHO; Spalart-Allmaras; ZEUS; Gmsh; FLICA-OVAP; ReALE; NIRVANA; CUDA Referenced in: 694 Publications
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### Referenced by 1,128 Authors

 30 Berthon, Christophe 19 Dumbser, Michael 15 Toro, Eleuterio Francisco 14 Balsara, Dinshaw S. 14 Chalons, Christophe 14 Saurel, Richard 11 Turpault, Rodolphe 10 Parés Madroñal, Carlos 9 Kurganov, Alexander 9 Nkonga, Boniface 9 Shu, Chi-Wang 8 Castro, Manuel J. 8 Coquel, Frédéric 7 Capdeville, Guy 7 Dubroca, Bruno 7 Foucher, Françoise 7 Liu, Tiegang 6 Bouchut, François 6 Chen, Shusheng 6 Titarev, Vladimir A. 6 van Leer, Bram 6 Yan, Chao 5 Adams, Nikolaus A. 5 Boscheri, Walter 5 Clain, Stéphane Louis 5 Cueto-Felgueroso, Luis 5 Godlewski, Edwige 5 Groth, Clinton P. T. 5 Guinot, Vincent 5 Harten, Ami 5 Helluy, Philippe 5 Hérard, Jean-Marc 5 Hu, Lijun 5 Kapen, Pascalin Tiam 5 Khoo, Boo Cheong 5 Kitamura, Keiichi 5 McDonald, James G. 5 Munz, Claus-Dieter 5 Seguin, Nicolas 5 Shen, Zhijun 5 Shyue, Keh-Ming 5 Tian, Zhengyu 5 Vázquez-Cendón, María Elena 5 Xiao, Feng 5 Xie, Wenfeng 5 Xu, Kun 4 Abgrall, Rémi 4 Castro-Díaz, Manuel Jesús 4 Causon, D. M. 4 Chiapolino, Alexandre 4 Colominas, Ignasi 4 Dawson, Clint N. 4 Drikakis, Dimitris 4 Fernández-Nieto, Enrique Domingo 4 Font, José A. 4 Frank, Martin 4 Guisset, Sebastien 4 Ingram, David M. 4 Klingenberg, Christian 4 Kusch, Jonas 4 Le, Minh Hoang 4 LeFloch, Philippe Gerard 4 Liu, Kai 4 Mandal, Jadav Chandra 4 Mathis, Hélène 4 Michel-Dansac, Victor 4 Morales de Luna, Tomás 4 Navarrina, Fermín 4 Pelanti, Marica 4 Perthame, Benoît 4 Puertas, Jerónimo 4 Qamar, Shamsul 4 Raghurama Rao, S. V. 4 Raviart, Pierre-Arnaud 4 Rosatti, Giorgio 4 San, Omer 4 Waagan, Knut 4 Wendroff, Burton 4 Xie, Wenjia 3 Adami, Stefan 3 Audit, Edouard 3 Bermúdez, Alfredo 3 Blachère, Florian 3 Brull, Stephane 3 Cai, Zhenning 3 Cancès, Clément 3 Celledoni, Elena 3 Chakravarthy, Sukumar R. 3 Chandrashekar, Praveen 3 Chertock, Alina E. 3 Daude, Frédéric 3 Davis, Stephen F. 3 De Vuyst, Florian 3 Deng, Xi 3 Dong, Jian 3 Duran, Arnaud 3 Duvigneau, Régis 3 Fleischmann, Nico 3 Fridrich, David 3 Galon, Pascal ...and 1,028 more Authors
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### Referenced in 100 Serials

 230 Journal of Computational Physics 89 Computers and Fluids 26 Journal of Scientific Computing 21 Computer Methods in Applied Mechanics and Engineering 16 International Journal for Numerical Methods in Fluids 16 SIAM Journal on Scientific Computing 14 Communications in Computational Physics 13 Computers & Mathematics with Applications 13 Applied Numerical Mathematics 12 Mathematics of Computation 12 Journal of Computational and Applied Mathematics 11 Journal of Fluid Mechanics 11 SIAM Journal on Numerical Analysis 11 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 10 Applied Mathematics and Computation 8 Numerische Mathematik 8 International Journal of Computational Fluid Dynamics 7 Shock Waves 7 Applied Mathematical Modelling 6 International Journal on Finite Volumes 5 Numerical Methods for Partial Differential Equations 5 European Journal of Mechanics. B. Fluids 5 Archives of Computational Methods in Engineering 4 Classical and Quantum Gravity 4 International Journal for Numerical Methods in Engineering 4 Computational Mechanics 4 Computational Mathematics and Mathematical Physics 4 Continuum Mechanics and Thermodynamics 4 M2AN. Mathematical Modelling and Numerical Analysis. ESAIM, European Series in Applied and Industrial Mathematics 4 Communications in Nonlinear Science and Numerical Simulation 4 Advances in Applied Mathematics and Mechanics 4 SMAI Journal of Computational Mathematics 3 Computational Mathematics and Modeling 3 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings 3 Living Reviews in Relativity 2 Computer Physics Communications 2 ZAMP. Zeitschrift für angewandte Mathematik und Physik 2 Zhurnal Vychislitel’noĭ Matematiki i Matematicheskoĭ Fiziki 2 Calcolo 2 Mathematics and Computers in Simulation 2 SIAM Journal on Scientific and Statistical Computing 2 Applied Mathematics Letters 2 Communications in Numerical Methods in Engineering 2 Advances in Computational Mathematics 2 Mathematical Problems in Engineering 2 Computational Geosciences 2 Communications in Mathematical Sciences 2 Applied Mathematical Sciences 2 Networks and Heterogeneous Media 2 SIAM/ASA Journal on Uncertainty Quantification 2 International Journal of Applied and Computational Mathematics 1 Communications on Pure and Applied Mathematics 1 Journal of Engineering Mathematics 1 Journal of Statistical Physics 1 Mathematical Methods in the Applied Sciences 1 Nonlinearity 1 Bulletin of the Polish Academy of Sciences. Technical Sciences 1 Kybernetika 1 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 1 Transactions of the American Mathematical Society 1 Journal of Computational Mathematics 1 Acta Mathematicae Applicatae Sinica. English Series 1 Mathematical and Computer Modelling 1 Japan Journal of Industrial and Applied Mathematics 1 M$$^3$$AS. Mathematical Models & Methods in Applied Sciences 1 SIAM Review 1 Mémoires de la Société Mathématique de France. Nouvelle Série 1 Philosophical Transactions of the Royal Society of London. Series A 1 Journal of Nonlinear Science 1 Computational Optimization and Applications 1 Physics of Fluids 1 Annales Mathématiques Blaise Pascal 1 International Journal of Numerical Methods for Heat & Fluid Flow 1 Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 1 Flow, Turbulence and Combustion 1 Lobachevskii Journal of Mathematics 1 Nonlinear Analysis. Real World Applications 1 Matematicheskoe Modelirovanie 1 Journal of Applied Mathematics 1 Communications in Information and Systems 1 Bulletin of the Brazilian Mathematical Society. New Series 1 Multiscale Modeling & Simulation 1 Acta Numerica 1 Springer Series in Computational Mathematics 1 Annales Universitatis Mariae Curie-Skłodowska. Sectio A1. Informatica 1 International Journal of Mathematical Analysis (Ruse) 1 SIAM Journal on Imaging Sciences 1 Kinetic and Related Models 1 Numerical Mathematics: Theory, Methods and Applications 1 Confluentes Mathematici 1 GEM - International Journal on Geomathematics 1 International Journal of Advances in Engineering Sciences and Applied Mathematics 1 S$$\vec{\text{e}}$$MA Journal 1 Journal of Computational Methods in Physics 1 AMM. Applied Mathematics and Mechanics. (English Edition) 1 European Series in Applied and Industrial Mathematics (ESAIM): Proceedings and Surveys 1 Proceedings of the Royal Society of London. A. Mathematical, Physical and Engineering Sciences 1 Frontiers in Mathematics 1 Communications on Applied Mathematics and Computation 1 Journal of Computational and Theoretical Transport
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### Referenced in 32 Fields

 558 Fluid mechanics (76-XX) 382 Numerical analysis (65-XX) 210 Partial differential equations (35-XX) 31 Geophysics (86-XX) 25 Classical thermodynamics, heat transfer (80-XX) 21 Mechanics of deformable solids (74-XX) 21 Statistical mechanics, structure of matter (82-XX) 17 Astronomy and astrophysics (85-XX) 10 Relativity and gravitational theory (83-XX) 9 Operations research, mathematical programming (90-XX) 8 Biology and other natural sciences (92-XX) 6 Calculus of variations and optimal control; optimization (49-XX) 5 Computer science (68-XX) 4 Dynamical systems and ergodic theory (37-XX) 4 Probability theory and stochastic processes (60-XX) 3 Optics, electromagnetic theory (78-XX) 2 Linear and multilinear algebra; matrix theory (15-XX) 2 Integral equations (45-XX) 2 Mechanics of particles and systems (70-XX) 1 General and overarching topics; collections (00-XX) 1 History and biography (01-XX) 1 Functions of a complex variable (30-XX) 1 Potential theory (31-XX) 1 Ordinary differential equations (34-XX) 1 Difference and functional equations (39-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Operator theory (47-XX) 1 Differential geometry (53-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Quantum theory (81-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX)