{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T05:00:45Z","timestamp":1776142845448,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2015,11,13]],"date-time":"2015-11-13T00:00:00Z","timestamp":1447372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Based on the lattice Boltzmann method (LBM), the lattice Boltzmann flux solver (LBFS), which combines the advantages of conventional Navier\u2013Stokes solvers and lattice Boltzmann solvers, was proposed recently. Specifically, LBFS applies the finite volume method to solve the macroscopic governing equations which provide solutions for macroscopic flow variables at cell centers. In the meantime, numerical fluxes at each cell interface are evaluated by local reconstruction of LBM solution. In other words, in LBFS, LBM is only locally applied at the cell interface for one streaming step. This is quite different from the conventional LBM, which is globally applied in the whole flow domain. This paper shows three different versions of LBFS respectively for isothermal, thermal and compressible flows and their relationships with the standard LBM. In particular, the performance of isothermal LBFS in terms of accuracy, efficiency and stability is investigated by comparing it with the standard LBM. The thermal LBFS is simplified by using the D2Q4 lattice velocity model and its performance is examined by its application to simulate natural convection with high Rayleigh numbers. It is demonstrated that the compressible LBFS can be effectively used to simulate both inviscid and viscous flows by incorporating non-equilibrium effects into the process for inviscid flux reconstruction. Several numerical examples, including lid-driven cavity flow, natural convection in a square cavity at Rayleigh numbers of 107 and 108 and transonic flow around a staggered-biplane configuration, are tested on structured or unstructured grids to examine the performance of three LBFS versions. Good agreements have been achieved with the published data, which validates the capability of LBFS in simulating a variety of flow problems.<\/jats:p>","DOI":"10.3390\/e17117713","type":"journal-article","created":{"date-parts":[[2015,11,16]],"date-time":"2015-11-16T05:40:47Z","timestamp":1447652447000},"page":"7713-7735","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":53,"title":["From Lattice Boltzmann Method to Lattice Boltzmann Flux Solver"],"prefix":"10.3390","volume":"17","author":[{"given":"Yan","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liming","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing 210016, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0825-0883","authenticated-orcid":false,"given":"Chang","family":"Shu","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2332","DOI":"10.1103\/PhysRevLett.61.2332","article-title":"Use of the Boltzmann Equation to Simulate Lattice-Gas Automata","volume":"61","author":"McNamara","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1209\/0295-5075\/9\/4\/008","article-title":"Lattice gas-dynamics with enhanced collisions","volume":"9","author":"Higuera","year":"1989","journal-title":"Europhys. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3776","DOI":"10.1103\/PhysRevLett.67.3776","article-title":"Lattice Boltzmann model for simulation of magnetohydrodynamics","volume":"67","author":"Chen","year":"1991","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1209\/0295-5075\/17\/6\/001","article-title":"Lattice BGK models for Navier\u2013Stokes equation","volume":"17","author":"Qian","year":"1992","journal-title":"Europhys. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1006\/jcph.1998.5984","article-title":"On the Finite Difference-Based Lattice Boltzmann Method in Curvilinear Coordinates","volume":"143","author":"Mei","year":"1998","journal-title":"J. Comput. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.jcp.2014.02.030","article-title":"Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates","volume":"267","author":"Hejranfar","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.compfluid.2013.02.018","article-title":"Finite-volume lattice Boltzmann modeling of thermal transport in nanofluids","volume":"77","author":"Zarghami","year":"2013","journal-title":"Comput. Fluids"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.jcp.2014.09.035","article-title":"Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio","volume":"280","author":"Wang","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.compfluid.2014.02.006","article-title":"Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows","volume":"94","author":"Wang","year":"2014","journal-title":"Comput. Fluids"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"436","DOI":"10.4208\/aamm.2014.4.s2","article-title":"Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows","volume":"6","author":"Shu","year":"2014","journal-title":"Adv. Appl. Math. Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.compfluid.2013.03.020","article-title":"A moment conservation-based non-free parameter compressible lattice Boltzmann model and its application for flux evaluation at cell interface","volume":"79","author":"Yang","year":"2013","journal-title":"Comput. Fluids"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"288","DOI":"10.1006\/jcph.2000.6616","article-title":"Lattice BGK Model for Incompressible Navier\u2013Stokes Equation","volume":"165","author":"Guo","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1815","DOI":"10.1103\/PhysRevE.47.1815","article-title":"Lattice Boltzmann model for simulating flows with multiple phases and components","volume":"47","author":"Shan","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.jcp.2004.12.001","article-title":"A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio","volume":"206","author":"Lee","year":"2005","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1006\/jcph.1999.6257","article-title":"A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh\u2013Taylor Instability","volume":"152","author":"He","year":"1999","journal-title":"J. Comput. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"056705","DOI":"10.1103\/PhysRevE.76.056705","article-title":"Coupled double-distribution-function lattice Boltzmann method for the compressible Navier\u2013Stokes equations","volume":"76","author":"Li","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Chen, F., Xu, A., Zhang, G., Li, Y., and Succi, S. (2010). Multiple-relaxation-time lattice Boltzmann approach to compressible flows with flexible specific-heat ratio and Prandtl number. Europhys. Lett., 90.","DOI":"10.1209\/0295-5075\/90\/54003"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4884","DOI":"10.1016\/j.physa.2013.06.021","article-title":"Three-dimensional finite-difference lattice Boltzmann model and its application to inviscid compressible flows with shock waves","volume":"392","author":"He","year":"2013","journal-title":"Physica A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1023\/A:1014523007427","article-title":"Lattice-Boltzmann Simulations of Fluid Flows in MEMS","volume":"107","author":"Nie","year":"2002","journal-title":"J. Stat. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2299","DOI":"10.1063\/1.1483841","article-title":"Application of lattice Boltzmann method to simulate microchannel flows","volume":"14","author":"Lim","year":"2002","journal-title":"Phys. Fluids"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1016\/j.jcp.2010.10.023","article-title":"Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows","volume":"230","author":"Meng","year":"2011","journal-title":"J. Comput. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6546","DOI":"10.1103\/PhysRevE.61.6546","article-title":"Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability","volume":"61","author":"Lallemand","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1209\/epl\/i2003-10307-8","article-title":"A lattice Boltzmann BGK model for simulation of micro flows","volume":"67","author":"Niu","year":"2004","journal-title":"Europhys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1016\/j.camwa.2014.01.006","article-title":"A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating a flexible filament in an incompressible flow","volume":"67","author":"Yuan","year":"2014","journal-title":"Comput. Math. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"307","DOI":"10.4208\/aamm.2013.m409","article-title":"A Numerical Study of Jet Propulsion of an Oblate Jelly fish Using a Momentum Exchange-Based Immersed Boundary-Lattice Boltzmann Method","volume":"6","author":"Yuan","year":"2014","journal-title":"Adv. Appl. Math. Mech."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1504\/PCFD.2005.005820","article-title":"Recent advances of Lattice Boltzmann techniques on unstructured grids","volume":"5","author":"Ubertini","year":"2005","journal-title":"Prog. Comput. Fluid Dyn. Int. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1002\/fld.1018","article-title":"Unstructured lattice Boltzmann method in three dimensions","volume":"49","author":"Rossi","year":"2005","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1016\/0017-9310(72)90054-3","article-title":"A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows","volume":"15","author":"Patankar","year":"1972","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Sun, D.L., Qu, Z.G., He, Y.L., and Tao, W.Q. (2008). An efficient segregated algorithm for incompressible fluid flow and heat transfer problems-IDEAL (Inner Doubly Iterative Efficient Algorithm for Linked Equations) Part I: Mathematical formulation and solution procedure. Numer. Heat Transf. B, 53.","DOI":"10.1080\/10407790701632543"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.compfluid.2004.12.004","article-title":"The 2D lid-driven cavity problem revisited","volume":"35","author":"Bruneau","year":"2006","journal-title":"Comput. Fluids"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1002\/fld.2013","article-title":"A streamfunction\u2013velocity approach for 2D transient incompressible viscous flows","volume":"62","author":"Kalita","year":"2010","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.compfluid.2012.01.018","article-title":"Multi relaxation time lattice Boltzmann simulations of transition in deep 2D lid driven cavity using GPU","volume":"80","author":"Lin","year":"2013","journal-title":"Comput. Fluids"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1863","DOI":"10.1016\/j.camwa.2013.02.020","article-title":"Filter-matrix lattice Boltzmann simulation of lid-driven deep-cavity flows, Part I\u2014Steady flows","volume":"65","author":"Zhuo","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1016\/j.camwa.2013.03.002","article-title":"Filter-matrix lattice Boltzmann simulation of lid-driven deep-cavity flows, Part II\u2014Flow bifurcation","volume":"65","author":"Zhuo","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/j.jcp.2014.06.047","article-title":"Lattice-Boltzmann simulations of the thermally driven 2D square cavity at high Rayleigh numbers","volume":"275","author":"Contrino","year":"2014","journal-title":"J. Comput. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1016\/j.ijheatmasstransfer.2011.01.004","article-title":"Evaluation of the coupling scheme of FVM and LBM for fluid flows around complex geometries","volume":"54","author":"Luan","year":"2011","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1002\/fld.337","article-title":"A coupled lattice BGK model for the Boussinesq equations","volume":"39","author":"Guo","year":"2002","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.jcp.2013.07.034","article-title":"Coupled numerical approach combining finite volume and lattice Boltzmann methods for multi-scale multi-physicochemical processes","volume":"255","author":"Chen","year":"2013","journal-title":"J. Comput. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3834","DOI":"10.1016\/j.ijheatmasstransfer.2012.02.020","article-title":"Coupling between finite volume method and lattice Boltzmann method and its application to fluid flow and mass transport in proton exchange membrane fuel cell","volume":"55","author":"Chen","year":"2012","journal-title":"Int. J. Heat Mass Transf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/j.jcp.2013.08.025","article-title":"Circular function-based gas-kinetic scheme for simulation of inviscid compressible flows","volume":"255","author":"Yang","year":"2013","journal-title":"J. Comput. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.jcp.2015.03.058","article-title":"A three-dimensional explicit sphere function-based gas-kinetic flux solver for simulation of inviscid compressible flows","volume":"295","author":"Yang","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"026701","DOI":"10.1103\/PhysRevE.68.026701","article-title":"Simplified thermal lattice Boltzmann model for incompressible thermal flows","volume":"68","author":"Peng","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0045-7930(91)90025-D","article-title":"Accurate solutions to the square thermally driven cavity at high Rayleigh number","volume":"20","year":"1991","journal-title":"Comput. Fluids"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1006\/jcph.2000.6596","article-title":"A High-Resolution Procedure for Euler and Navier\u2013Stokes Computations on Unstructured Grids","volume":"164","author":"Jawahar","year":"2000","journal-title":"J. Comput. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1006\/jcph.1996.0004","article-title":"Stable Conservative Multidomain Treatments for Implicit Euler Solvers","volume":"123","author":"Lerat","year":"1996","journal-title":"J. Comput. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/11\/7713\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:52:02Z","timestamp":1760215922000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/11\/7713"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,13]]},"references-count":45,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2015,11]]}},"alternative-id":["e17117713"],"URL":"https:\/\/doi.org\/10.3390\/e17117713","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,11,13]]}}}