{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:01:27Z","timestamp":1766268087505,"version":"3.40.3"},"reference-count":66,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,3,8]],"date-time":"2025-03-08T00:00:00Z","timestamp":1741392000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,3,8]],"date-time":"2025-03-08T00:00:00Z","timestamp":1741392000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12072127","51836003","123B2018"],"award-info":[{"award-number":["12072127","51836003","123B2018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1007\/s10915-025-02844-x","type":"journal-article","created":{"date-parts":[[2025,3,8]],"date-time":"2025-03-08T12:58:38Z","timestamp":1741438718000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Cole\u2013Hopf Transformation Based Fourth-Order Multiple-Relaxation-Time Lattice Boltzmann Model for the Coupled Burgers\u2019 Equations"],"prefix":"10.1007","volume":"103","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-8041-3915","authenticated-orcid":false,"given":"Ying","family":"Chen","sequence":"first","affiliation":[]},{"given":"Xi","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Zhenhua","family":"Chai","sequence":"additional","affiliation":[]},{"given":"Baochang","family":"Shi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,3,8]]},"reference":[{"issue":"1","key":"2844_CR1","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1134\/S1990478915010020","volume":"9","author":"YE Anikonov","year":"2015","unstructured":"Anikonov, Y.E., Ayupova, N.: The Hopf\u2013Cole transformation and multidimensional representations of solutions to evolution equations. J. Appl. Ind. Math. 9(1), 11 (2015)","journal-title":"J. Appl. Ind. Math."},{"issue":"8","key":"2844_CR2","first-page":"897","volume":"1","author":"AR Bahadir","year":"1999","unstructured":"Bahadir, A.R.: Numerical solution for one-dimensional Burgers\u2019 equation using a fully implicit finite-difference method. Int. J. Appl. Math. Comput. Sci. 1(8), 897\u2013910 (1999)","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"issue":"3","key":"2844_CR3","doi-asserted-by":"publisher","first-page":"1225","DOI":"10.1051\/m2an\/2023008","volume":"57","author":"T Bellotti","year":"2023","unstructured":"Bellotti, T.: Truncation errors and modified equations for the lattice Boltzmann method via the corresponding Finite Difference schemes. ESAIM: Math. Model. Numer. Anal. 57(3), 1225\u20131255 (2023)","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"key":"2844_CR4","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/j.camwa.2024.09.028","volume":"174","author":"T Bellotti","year":"2024","unstructured":"Bellotti, T.: The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes. Comput. Math. Appl. 174, 397\u2013416 (2024)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"2844_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00211-022-01302-2","volume":"152","author":"T Bellotti","year":"2022","unstructured":"Bellotti, T., Graille, B., Massot, M.: Finite difference formulation of any lattice Boltzmann scheme. Numer. Math. 152(1), 1\u201340 (2022)","journal-title":"Numer. Math."},{"key":"2844_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2024.106410","volume":"284","author":"BM Boghosian","year":"2024","unstructured":"Boghosian, B.M., Dubois, F., Lallemand, P.: Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations. Comput. Fluids 284, 106410 (2024)","journal-title":"Comput. Fluids"},{"issue":"1821","key":"2844_CR7","doi-asserted-by":"publisher","first-page":"1691","DOI":"10.1098\/rsta.2004.1410","volume":"362","author":"BM Boghosian","year":"2004","unstructured":"Boghosian, B.M., Love, P., Yepez, J.: Entropic lattice Boltzmann model for Burgers\u2019s equation. Philos. Trans. Royal Soc. A 362(1821), 1691\u20131701 (2004)","journal-title":"Philos. Trans. Royal Soc. A"},{"key":"2844_CR8","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1017\/jfm.2012.288","volume":"707","author":"R Bonhomme","year":"2012","unstructured":"Bonhomme, R., Magnaudet, J., Duval, F., Piar, B.: Inertial dynamics of air bubbles crossing a horizontal fluid-fluid interface. J. Fluid Mech. 707, 405\u2013443 (2012)","journal-title":"J. Fluid Mech."},{"issue":"3","key":"2844_CR9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.033109","volume":"93","author":"M Buzzicotti","year":"2016","unstructured":"Buzzicotti, M., Biferale, L., Frisch, U., Ray, S.S.: Intermittency in fractal Fourier hydrodynamics: lessons from the Burgers equation. Phys. Rev. E 93(3), 033109 (2016)","journal-title":"Phys. Rev. E"},{"issue":"1","key":"2844_CR10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.97.013304","volume":"97","author":"Z Chai","year":"2018","unstructured":"Chai, Z., He, N., Guo, Z., Shi, B.: Lattice Boltzmann model for high-order nonlinear partial differential equations. Phys. Rev. E 97(1), 013304 (2018)","journal-title":"Phys. Rev. E"},{"issue":"4","key":"2844_CR11","doi-asserted-by":"publisher","first-page":"874","DOI":"10.1016\/j.chaos.2006.07.023","volume":"36","author":"Z Chai","year":"2008","unstructured":"Chai, Z., Shi, B., Zheng, L.: A unified lattice Boltzmann model for some nonlinear partial differential equations. Chaos Solitons & Fractals 36(4), 874\u2013882 (2008)","journal-title":"Chaos Solitons & Fractals"},{"issue":"2","key":"2844_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.102.023306","volume":"102","author":"Z Chai","year":"2020","unstructured":"Chai, Z., Shi, B., et al.: Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: modeling, analysis, and elements. Phys. Rev. E 102(2), 023306 (2020)","journal-title":"Phys. Rev. E"},{"issue":"1","key":"2844_CR13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.108.015304","volume":"108","author":"Z Chai","year":"2023","unstructured":"Chai, Z., Yuan, X., Shi, B.: Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: general equilibrium and some important issues. Phys. Rev. E 108(1), 015304 (2023)","journal-title":"Phys. Rev. E"},{"key":"2844_CR14","volume-title":"The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases","author":"S Chapman","year":"1990","unstructured":"Chapman, S., Cowling, T.G.: The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge University Press, Cambridge (1990)"},{"issue":"5","key":"2844_CR15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.107.055305","volume":"107","author":"Y Chen","year":"2023","unstructured":"Chen, Y., Chai, Z., Shi, B.: Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations. Phys. Rev. E 107(5), 055305 (2023)","journal-title":"Phys. Rev. E"},{"key":"2844_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2024.113045","volume":"509","author":"Y Chen","year":"2024","unstructured":"Chen, Y., Chai, Z., Shi, B.: A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and equivalent macroscopic finite-difference scheme for two-dimensional diffusion equations. J. Comput. Phys. 509, 113045 (2024)","journal-title":"J. Comput. Phys."},{"key":"2844_CR17","unstructured":"Chen, Y., Chai, Z., Shi, B.: A unified fourth-order Bhatnagar-Gross-Krook lattice Boltzmann model for high-dimensional linear hyperbolic equations. (2024). arXiv preprint arXiv:2410.13165"},{"issue":"6","key":"2844_CR18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.109.065305","volume":"109","author":"Y Chen","year":"2024","unstructured":"Chen, Y., Liu, X., Chai, Z., Shi, B.: Macroscopic finite-difference scheme and modified equations of the general propagation multiple-relaxation-time lattice boltzmann model. Phys. Rev. E 109(6), 065305 (2024)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"2844_CR19","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1090\/qam\/42889","volume":"9","author":"JD Cole","year":"1951","unstructured":"Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9(3), 225\u2013236 (1951)","journal-title":"Q. Appl. Math."},{"issue":"1","key":"2844_CR20","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1146\/annurev.fl.11.010179.000303","volume":"11","author":"DG Crighton","year":"1979","unstructured":"Crighton, D.G.: Model equations of nonlinear acoustics. Annu. Rev. Fluid Mech. 11(1), 11\u201333 (1979)","journal-title":"Annu. Rev. Fluid Mech."},{"issue":"4","key":"2844_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.043311","volume":"93","author":"S Cui","year":"2016","unstructured":"Cui, S., Hong, N., Shi, B., Chai, Z.: Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations. Phys. Rev. E 93(4), 043311 (2016)","journal-title":"Phys. Rev. E"},{"issue":"1792","key":"2844_CR22","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1098\/rsta.2001.0955","volume":"360","author":"D d\u2019Humi\u00e8res","year":"2002","unstructured":"d\u2019Humi\u00e8res, D.: Multiple-relaxation-time lattice Boltzmann models in three dimensions. Philos. Trans. Royal Soc. A 360(1792), 437\u2013451 (2002)","journal-title":"Philos. Trans. Royal Soc. A"},{"issue":"1","key":"2844_CR23","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1016\/j.cam.2006.08.002","volume":"206","author":"Y Duan","year":"2007","unstructured":"Duan, Y., Liu, R.: Lattice Boltzmann model for two-dimensional unsteady Burgers\u2019 equation. J. Comput. Appl. Math. 206(1), 432\u2013439 (2007)","journal-title":"J. Comput. Appl. Math."},{"issue":"7","key":"2844_CR24","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1016\/j.camwa.2007.08.003","volume":"55","author":"F Dubois","year":"2008","unstructured":"Dubois, F.: Equivalent partial differential equations of a lattice Boltzmann scheme. Comput. Math. Appl. 55(7), 1441\u20131449 (2008)","journal-title":"Comput. Math. Appl."},{"issue":"1\u20132","key":"2844_CR25","first-page":"221","volume":"23","author":"F Dubois","year":"2009","unstructured":"Dubois, F.: Third order equivalent equation of lattice Boltzmann scheme. Discret. Contin. Dyn. Syst. 23(1\u20132), 221\u2013248 (2009)","journal-title":"Discret. Contin. Dyn. Syst."},{"issue":"4","key":"2844_CR26","first-page":"297","volume":"127","author":"F Dubois","year":"2022","unstructured":"Dubois, F.: Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes. Asymptot. Anal. 127(4), 297\u2013337 (2022)","journal-title":"Asymptot. Anal."},{"issue":"5","key":"2844_CR27","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1016\/j.camwa.2009.02.008","volume":"58","author":"D d\u2019Humi\u00e8res","year":"2009","unstructured":"d\u2019Humi\u00e8res, D., Ginzburg, I.: Viscosity independent numerical errors for lattice Boltzmann models: From recurrence equations to \u201cmagic\u2019\u2019 collision numbers. Comput. Math. Appl. 58(5), 823\u2013840 (2009)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"2844_CR28","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1137\/0917052","volume":"17","author":"BH Elton","year":"1996","unstructured":"Elton, B.H.: Comparisons of lattice Boltzmann and finite difference methods for a two-dimensional viscous Burgers equation. SIAM J. Sci. Comput. 17(4), 783\u2013813 (1996)","journal-title":"SIAM J. Sci. Comput."},{"key":"2844_CR29","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/j.apm.2016.12.018","volume":"45","author":"Q Gao","year":"2017","unstructured":"Gao, Q., Zhou, M.Y.: An analytical solution for two and three dimensional nonlinear Burgers\u2019 equation. Appl. Math. Model. 45, 255\u2013270 (2017)","journal-title":"Appl. Math. Model."},{"issue":"5","key":"2844_CR30","doi-asserted-by":"publisher","first-page":"1439","DOI":"10.4208\/cicp.211210.280611a","volume":"11","author":"I Ginzburg","year":"2012","unstructured":"Ginzburg, I.: Truncation errors, exact and heuristic stability analysis of two-relaxation-times lattice Boltzmann schemes for anisotropic advection-diffusion equation. Commun. Comput. Phys. 11(5), 1439\u20131502 (2012)","journal-title":"Commun. Comput. Phys."},{"key":"2844_CR31","doi-asserted-by":"publisher","DOI":"10.1142\/8806","volume-title":"Lattice Boltzmann method and its application in engineering","author":"Z Guo","year":"2013","unstructured":"Guo, Z., Shu, C.: Lattice Boltzmann method and its application in engineering, vol. 3. World Scientific, Singapore (2013)"},{"issue":"2","key":"2844_CR32","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1016\/j.jcp.2003.08.012","volume":"193","author":"DJ Holdych","year":"2004","unstructured":"Holdych, D.J., Noble, D.R., Georgiadis, J.G., Buckius, R.O.: Truncation error analysis of lattice Boltzmann methods. J. Comput. Phys. 193(2), 595\u2013619 (2004)","journal-title":"J. Comput. Phys."},{"issue":"2\u20134","key":"2844_CR33","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1016\/j.physa.2004.10.041","volume":"350","author":"Z Horii","year":"2005","unstructured":"Horii, Z.: Mass transport theory for the Toda lattices, dispersive and dissipative. Phys. A 350(2\u20134), 349\u2013378 (2005)","journal-title":"Phys. A"},{"issue":"1","key":"2844_CR34","first-page":"1","volume":"5","author":"E Ikenberry","year":"1956","unstructured":"Ikenberry, E., Truesdell, C.: On the pressures and the flux of energy in a gas according to Maxwell\u2019s kinetic theory, I. J. Ration. Mech. Anal. 5(1), 1\u201354 (1956)","journal-title":"J. Ration. Mech. Anal."},{"key":"2844_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2007.04.002","volume":"447","author":"B J\u00e9r\u00e9mie","year":"2007","unstructured":"J\u00e9r\u00e9mie, B., Konstantin, K.: Burgers turbulence. Phys. Rep. 447, 1\u201366 (2007)","journal-title":"Phys. Rep."},{"key":"2844_CR36","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1016\/j.physa.2013.10.030","volume":"395","author":"H Lai","year":"2014","unstructured":"Lai, H., Ma, C.: A new lattice Boltzmann model for solving the coupled viscous Burgers\u2019 equation. Phys. A 395, 445\u2013457 (2014)","journal-title":"Phys. A"},{"issue":"6","key":"2844_CR37","doi-asserted-by":"publisher","first-page":"6546","DOI":"10.1103\/PhysRevE.61.6546","volume":"61","author":"P Lallemand","year":"2000","unstructured":"Lallemand, P., Luo, L.S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61(6), 6546 (2000)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"2844_CR38","doi-asserted-by":"publisher","first-page":"864","DOI":"10.1016\/j.camwa.2017.10.013","volume":"75","author":"Q Li","year":"2018","unstructured":"Li, Q., Chai, Z., Shi, B.: Lattice Boltzmann models for two-dimensional coupled Burgers\u2019 equations. Comput. Appl. Math. 75(3), 864\u2013875 (2018)","journal-title":"Comput. Appl. Math."},{"issue":"2","key":"2844_CR39","first-page":"755","volume":"206","author":"WY Liao","year":"2008","unstructured":"Liao, W.Y.: An implicit fourth-order compact finite difference scheme for one-dimensional Burgers\u2019 equation. Appl. Math. Comput. 206(2), 755\u2013764 (2008)","journal-title":"Appl. Math. Comput."},{"issue":"5","key":"2844_CR40","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1002\/fld.2163","volume":"64","author":"WY Liao","year":"2010","unstructured":"Liao, W.Y.: A fourth-order finite-difference method for solving the system of two-dimensional Burgers\u2019 equations. Int. J. Numer. Methods Fluids 64(5), 565\u2013590 (2010)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"2844_CR41","unstructured":"Lighthill, M.J.: Viscosity effects in sound waves of finite amplitude. In: Batchelor, G.K., Davies, R.M. (eds.) Surveys in Mechanics. Cambridge University Press, London (1956)"},{"issue":"1","key":"2844_CR42","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.104.015312","volume":"104","author":"Y Lin","year":"2021","unstructured":"Lin, Y., Hong, N., Shi, B., Chai, Z.: Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations. Phys. Rev. E 104(1), 015312 (2021)","journal-title":"Phys. Rev. E"},{"issue":"5","key":"2844_CR43","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.83.056710","volume":"83","author":"LS Luo","year":"2011","unstructured":"Luo, L.S., Liao, W., Chen, X., Peng, Y., Zhang, W., et al.: Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations. Phys. Rev. E 83(5), 056710 (2011)","journal-title":"Phys. Rev. E"},{"key":"2844_CR44","doi-asserted-by":"publisher","unstructured":"Mer, Y., Fengqun, Z., Chong, G.: The Sinc-Galerkin methods of the Burgers\u2019 equation based on the Hopf-Cole transformation. Chin. J. Comput. Mech. 6, 807\u2013812 (2019). https:\/\/doi.org\/10.7511\/jslx20181024001","DOI":"10.7511\/jslx20181024001"},{"issue":"6","key":"2844_CR45","doi-asserted-by":"publisher","first-page":"450","DOI":"10.1080\/15502280903111424","volume":"10","author":"R Mittal","year":"2009","unstructured":"Mittal, R., Jiwari, R.: Differential quadrature method for two-dimensional Burgers\u2019 equations. Int. J. Comput. Methods Eng. Sci. Mech. 10(6), 450\u2013459 (2009)","journal-title":"Int. J. Comput. Methods Eng. Sci. Mech."},{"key":"2844_CR46","doi-asserted-by":"publisher","DOI":"10.1007\/s12591-019-00468-w","author":"V Mukundan","year":"2019","unstructured":"Mukundan, V., Awasthi, A., Aswin, V.: Multistep methods for the numerical simulation of two-dimensional Burgers\u2019 equation. Differ. Equ. Dyn. Syst. (2019). https:\/\/doi.org\/10.1007\/s12591-019-00468-w","journal-title":"Differ. Equ. Dyn. Syst."},{"issue":"9","key":"2844_CR47","doi-asserted-by":"publisher","first-page":"1331","DOI":"10.1088\/0034-4885\/65\/9\/203","volume":"65","author":"T Nagatani","year":"2002","unstructured":"Nagatani, T.: The physics of traffic jams. Rep. Prog. Phys. 65(9), 1331 (2002)","journal-title":"Rep. Prog. Phys."},{"issue":"8\u20139","key":"2844_CR48","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1016\/j.compfluid.2005.03.008","volume":"35","author":"C Pan","year":"2006","unstructured":"Pan, C., Luo, L.S., Miller, C.T.: An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids 35(8\u20139), 898\u2013909 (2006)","journal-title":"Comput. Fluids"},{"issue":"6","key":"2844_CR49","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1080\/10407790.2017.1326769","volume":"71","author":"D Pan","year":"2017","unstructured":"Pan, D.: A high-order finite volume method for solving one-dimensional convection and diffusion equations. Numer. Heat Transf. Part B: Fundam. 71(6), 533\u2013548 (2017)","journal-title":"Numer. Heat Transf. Part B: Fundam."},{"issue":"6","key":"2844_CR50","first-page":"2206","volume":"215","author":"K Pandey","year":"2009","unstructured":"Pandey, K., Verma, L., Verma, A.K.: On a finite difference scheme for Burgers\u2019 equation. Appl. Math. Comput. 215(6), 2206\u20132214 (2009)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"2844_CR51","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1088\/0253-6102\/69\/3\/329","volume":"69","author":"XT Qi","year":"2018","unstructured":"Qi, X.T., Shi, B.C., Chai, Z.H.: Cole\u2013Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers\u2019 Equation. Commun. Theor. Phys. 69(3), 329 (2018)","journal-title":"Commun. Theor. Phys."},{"issue":"2","key":"2844_CR52","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1002\/cpa.3160480202","volume":"48","author":"LG Reyna","year":"1995","unstructured":"Reyna, L.G., Ward, M.J.: On the exponentially slow motion of a viscous shock. Commun. Pure Appl. Math. 48(2), 79\u2013120 (1995)","journal-title":"Commun. Pure Appl. Math."},{"key":"2844_CR53","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.camwa.2023.01.017","volume":"134","author":"F Rong","year":"2023","unstructured":"Rong, F., Li, Q., Shi, B., Chai, Z.: A lattice Boltzmann model based on Cole\u2013Hopf transformation for N-dimensional coupled Burgers\u2019 equations. Comput. Math. Appl. 134, 101\u2013111 (2023)","journal-title":"Comput. Math. Appl."},{"key":"2844_CR54","doi-asserted-by":"crossref","unstructured":"Strikwerda, J.C.: Finite difference schemes and partial differential equations. SIAM (2004)","DOI":"10.1137\/1.9780898717938"},{"key":"2844_CR55","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198503989.001.0001","volume-title":"The lattice Boltzmann equation: for fluid dynamics and beyond","author":"S Succi","year":"2001","unstructured":"Succi, S.: The lattice Boltzmann equation: for fluid dynamics and beyond. Oxford University Press, Oxford (2001)"},{"key":"2844_CR56","doi-asserted-by":"publisher","first-page":"494","DOI":"10.1007\/s10955-010-0004-y","volume":"140","author":"S Suga","year":"2010","unstructured":"Suga, S.: An accurate multi-level finite difference scheme for 1D diffusion equations derived from the lattice Boltzmann method. J. Stat. Phys. 140, 494\u2013503 (2010)","journal-title":"J. Stat. Phys."},{"key":"2844_CR57","unstructured":"T., K., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G., Viggen, E.: The lattice Boltzmann method: principles and practice. Cham, Switzerland: Springer International Publishing AG (2016)"},{"issue":"1","key":"2844_CR58","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.physa.2005.09.031","volume":"362","author":"A Velivelli","year":"2006","unstructured":"Velivelli, A., Bryden, K.: Parallel performance and accuracy of lattice Boltzmann and traditional finite difference methods for solving the unsteady two-dimensional Burger\u2019s equation. Phys. A 362(1), 139\u2013145 (2006)","journal-title":"Phys. A"},{"key":"2844_CR59","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.74.056703","volume":"74","author":"AJ Wagner","year":"2006","unstructured":"Wagner, A.J.: Thermodynamic consistency of liquid-gas lattice Boltzmann simulations. Phys. Rev. E 74, 056703 (2006)","journal-title":"Phys. Rev. E"},{"issue":"3","key":"2844_CR60","doi-asserted-by":"publisher","first-page":"33","DOI":"10.26804\/capi.2019.03.01","volume":"2","author":"HL Wang","year":"2019","unstructured":"Wang, H.L., Yuan, X.L., Liang, H., Chai, Z.H., Shi, B.C.: A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity 2(3), 33\u201352 (2019)","journal-title":"Capillarity"},{"issue":"9","key":"2844_CR61","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/0021-9991(74)90011-4","volume":"145","author":"RF Warming","year":"1974","unstructured":"Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comput. Phys. 145(9), 159\u2013179 (1974)","journal-title":"J. Comput. Phys."},{"key":"2844_CR62","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.033310","volume":"93","author":"WA Yong","year":"2016","unstructured":"Yong, W.A., Zhao, W., Luo, L.S.: Theory of the lattice Boltzmann method: derivation of macroscopic equations via the Maxwell iteration. Phys. Rev. E 93, 033310 (2016)","journal-title":"Phys. Rev. E"},{"key":"2844_CR63","doi-asserted-by":"publisher","first-page":"13342","DOI":"10.1002\/mma.9255","volume":"46","author":"X Yu","year":"2023","unstructured":"Yu, X., Zhang, L., Hu, B., Hu, Y.: A multiple-relaxation-time lattice Boltzmann model for Burgers equation. Math. Methods Appl. Sci. 46, 13342 (2023)","journal-title":"Math. Methods Appl. Sci."},{"issue":"19\u201320","key":"2844_CR64","doi-asserted-by":"publisher","first-page":"4771","DOI":"10.1016\/j.physa.2008.04.002","volume":"387","author":"J Zhang","year":"2008","unstructured":"Zhang, J., Yan, G.: Lattice Boltzmann method for one and two-dimensional Burgers equation. Phys. A 387(19\u201320), 4771\u20134786 (2008)","journal-title":"Phys. A"},{"issue":"02","key":"2844_CR65","doi-asserted-by":"publisher","first-page":"120","DOI":"10.4236\/ajcm.2016.62013","volume":"6","author":"T Zhanlav","year":"2016","unstructured":"Zhanlav, T., Chuluunbaatar, O., Ulziibayar, V.: Higher-order numerical solution of two-dimensional coupled Burgers\u2019 equations. Am. J. Comput. Math. 6(02), 120\u2013129 (2016)","journal-title":"Am. J. Comput. Math."},{"issue":"8","key":"2844_CR66","doi-asserted-by":"publisher","first-page":"3279","DOI":"10.1016\/j.camwa.2011.08.044","volume":"62","author":"G Zhao","year":"2011","unstructured":"Zhao, G., Yu, X., Zhang, R.: The new numerical method for solving the system of two-dimensional Burgers\u2019 equations. Comput. Math. Appl. 62(8), 3279\u20133291 (2011)","journal-title":"Comput. Math. Appl."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02844-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02844-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02844-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T03:12:36Z","timestamp":1743477156000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02844-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,8]]},"references-count":66,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["2844"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02844-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2025,3,8]]},"assertion":[{"value":"7 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 December 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 February 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 March 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have not disclosed any competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"25"}}