{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T14:46:20Z","timestamp":1772635580325,"version":"3.50.1"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T00:00:00Z","timestamp":1769817600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T00:00:00Z","timestamp":1769817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["MPS-TSM-00007854"],"award-info":[{"award-number":["MPS-TSM-00007854"]}],"id":[{"id":"10.13039\/100000893","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":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We develop a conservative and positivity-preserving discontinuous Galerkin (DG) method for the population balance equation (PBE), which models the distribution of particle numbers across particle sizes due to growth, nucleation, aggregation, and breakage. To ensure number conservation in growth and mass conservation in aggregation and breakage, we design a DG scheme that applies standard treatment for growth and nucleation, and introduces a novel discretization for aggregation and breakage. The birth and death terms are discretized in a symmetric double-integral form, evaluated using a common refinement of the integration domain and carefully selected quadrature rules. Beyond conservation, we focus on preserving the positivity of the number density in aggregation\u2013breakage. Since local mass corresponds to the first moment, the classical Zhang\u2013Shu limiter, which preserves the zeroth moment (i.e., cell average), is not directly applicable. We address this by proving the positivity of the first moment on each cell and constructing a moment-conserving limiter that enforces nonnegativity across the domain. To the best of our knowledge, this is the first work to develop a positivity-preserving algorithm that conserves a prescribed moment. The numerical results verify the accuracy, conservation, and robustness of the proposed method.<\/jats:p>","DOI":"10.1007\/s10915-026-03197-9","type":"journal-article","created":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T02:27:18Z","timestamp":1769826438000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation"],"prefix":"10.1007","volume":"106","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7298-004X","authenticated-orcid":false,"given":"Ziyao","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanyang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong-Tao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,1,31]]},"reference":[{"key":"3197_CR1","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/S0168-9274(97)00056-1","volume":"25","author":"UM Ascher","year":"1997","unstructured":"Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151\u2013167 (1997)","journal-title":"Appl. Numer. Math."},{"key":"3197_CR2","doi-asserted-by":"publisher","first-page":"742","DOI":"10.1016\/j.ces.2008.05.006","volume":"64","author":"MM Attarakih","year":"2009","unstructured":"Attarakih, M.M., Drumm, C., Bart, H.J.: Solution of the population balance equation using the sectional quadrature method of moments (SQMOM). Chem. Eng. Sci. 64, 742\u2013752 (2009)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR3","doi-asserted-by":"publisher","first-page":"5165","DOI":"10.1029\/JC075i027p05165","volume":"75","author":"R Bleck","year":"1970","unstructured":"Bleck, R.: A fast, approximative method for integrating the stochastic coalescence equation. J. Geophys. Res. 75, 5165\u20135171 (1970)","journal-title":"J. Geophys. Res."},{"key":"3197_CR4","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1090\/S0025-5718-07-02054-6","volume":"77","author":"JP Bourgade","year":"2008","unstructured":"Bourgade, J.P., Filbet, F.: Convergence of a finite volume scheme for coagulation-fragmentation equations. Math. Comput. 77, 851\u2013882 (2008)","journal-title":"Math. Comput."},{"key":"3197_CR5","doi-asserted-by":"publisher","first-page":"398","DOI":"10.1016\/j.jcp.2006.09.012","volume":"223","author":"Y Cheng","year":"2007","unstructured":"Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations. J. Comput. Phys. 223, 398\u2013415 (2007)","journal-title":"J. Comput. Phys."},{"key":"3197_CR6","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1090\/S0025-5718-07-02045-5","volume":"77","author":"Y Cheng","year":"2008","unstructured":"Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for time-dependent partial differential equations with higher-order derivatives. Math. Comput. 77, 699\u2013730 (2008)","journal-title":"Math. Comput."},{"key":"3197_CR7","doi-asserted-by":"crossref","unstructured":"Chew, L.P.: Constrained Delaunay triangulations, Proceedings of the Third Annual Symposium on Computational Geometry, 215\u2013222 (1987)","DOI":"10.1145\/41958.41981"},{"key":"3197_CR8","first-page":"545","volume":"54","author":"B Cockburn","year":"1990","unstructured":"Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545\u2013581 (1990)","journal-title":"Math. Comput."},{"key":"3197_CR9","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.jcp.2003.09.007","volume":"194","author":"B Cockburn","year":"2004","unstructured":"Cockburn, B., Li, F., Shu, C.-W.: Locally divergence-free discontinuous Galerkin methods for the Maxwell equations. J. Comput. Phys. 194, 588\u2013610 (2004)","journal-title":"J. Comput. Phys."},{"key":"3197_CR10","first-page":"411","volume":"52","author":"B Cockburn","year":"1989","unstructured":"Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411\u2013435 (1989)","journal-title":"Math. Comput."},{"key":"3197_CR11","doi-asserted-by":"crossref","unstructured":"Cockburn, B., Shu, C.-W.: The Runge\u2013Kutta local projection-discontinuous-Galerkin finite element method for scalar conservation laws, ESAIM: Mathematical Modelling and Numerical Analysis, 25, 337\u2013361 (1991)","DOI":"10.1051\/m2an\/1991250303371"},{"key":"3197_CR12","doi-asserted-by":"publisher","first-page":"2440","DOI":"10.1137\/S0036142997316712","volume":"35","author":"B Cockburn","year":"1998","unstructured":"Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection\u2013diffusion systems. SIAM J. Numer. Anal. 35, 2440\u20132463 (1998)","journal-title":"SIAM J. Numer. Anal."},{"key":"3197_CR13","doi-asserted-by":"publisher","first-page":"2211","DOI":"10.1016\/S0009-2509(02)00112-4","volume":"57","author":"RB Diemer","year":"2002","unstructured":"Diemer, R.B., Olson, J.H.: A moment methodology for coagulation and breakage problems: Part 2\u2014moment models and distribution reconstruction. Chem. Eng. Sci. 57, 2211\u20132228 (2002)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR14","unstructured":"Drake, R.L.: A general mathematical survey of the coagulation equation, Topics in Current Aerosol Research, Pergamon Press, Oxford, 201\u2013376 (1972)"},{"key":"3197_CR15","doi-asserted-by":"publisher","first-page":"2004","DOI":"10.1137\/S1064827503429132","volume":"25","author":"F Filbet","year":"2004","unstructured":"Filbet, F., Lauren\u00e7ot, P.: Numerical simulation of the Smoluchowski coagulation equation. SIAM J. Sci. Comput. 25, 2004\u20132028 (2004)","journal-title":"SIAM J. Sci. Comput."},{"key":"3197_CR16","volume-title":"Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics","author":"SK Friedlander","year":"2000","unstructured":"Friedlander, S.K.: Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd edn. Oxford University Press, Oxford (2000)","edition":"2"},{"key":"3197_CR17","doi-asserted-by":"crossref","unstructured":"Gabriel, P., Tine, L.M.: High-order WENO scheme for polymerization-type equations, ESAIM: Proceedings, 30, 53-69 (2010)","DOI":"10.1051\/proc\/2010006"},{"key":"3197_CR18","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s10915-008-9239-z","volume":"38","author":"S Gottlieb","year":"2009","unstructured":"Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: High-order strong-stability-preserving time discretizations. J. Sci. Comput. 38, 251\u2013289 (2009)","journal-title":"J. Sci. Comput."},{"key":"3197_CR19","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1137\/S003614450036757X","volume":"43","author":"S Gottlieb","year":"2001","unstructured":"Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability preserving high-order time discretization methods. SIAM Rev. 43, 89\u2013112 (2001)","journal-title":"SIAM Rev."},{"key":"3197_CR20","doi-asserted-by":"publisher","first-page":"1975","DOI":"10.1016\/j.apm.2014.09.027","volume":"39","author":"A Hasseine","year":"2015","unstructured":"Hasseine, A., Bart, H.J.: Adomian decomposition method solution of population balance equations for aggregation, nucleation, growth and breakup processes. Appl. Math. Model. 39, 1975\u20131984 (2015)","journal-title":"Appl. Math. Model."},{"key":"3197_CR21","unstructured":"Hidy, G.M., Brock, J.R.: The dynamics of aerocolloidal systems, Pergamon (1970)"},{"key":"3197_CR22","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/j.apnum.2008.03.034","volume":"59","author":"DI Ketcheson","year":"2009","unstructured":"Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge-Kutta methods. Appl. Numer. Math. 59, 373\u2013392 (2009)","journal-title":"Appl. Numer. Math."},{"key":"3197_CR23","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1002\/bit.23321","volume":"109","author":"ME Kolewe","year":"2012","unstructured":"Kolewe, M.E., Roberts, S.C., Henson, M.A.: A population balance equation model of aggregation dynamics in Taxus suspension cell cultures. Biotechnol. Bioeng. 109, 472\u2013482 (2012)","journal-title":"Biotechnol. Bioeng."},{"key":"3197_CR24","first-page":"1311","volume":"51","author":"S Kumar","year":"1996","unstructured":"Kumar, S., Ramkrishna, D.: On the solution of population balance equations by discretization\u2014I. A fixed pivot technique, Chemical Engineering Science 51, 1311\u20131332 (1996)","journal-title":"A fixed pivot technique, Chemical Engineering Science"},{"key":"3197_CR25","first-page":"4659","volume":"52","author":"S Kumar","year":"1997","unstructured":"Kumar, S., Ramkrishna, D.: On the solution of population balance equations by discretization\u2014III. Nucleation, growth and aggregation of particles, Chemical Engineering Science 52, 4659\u20134679 (1997)","journal-title":"Nucleation, growth and aggregation of particles, Chemical Engineering Science"},{"key":"3197_CR26","doi-asserted-by":"publisher","first-page":"4253","DOI":"10.1016\/S0009-2509(02)00369-X","volume":"57","author":"PLC Lage","year":"2002","unstructured":"Lage, P.L.C.: Comments on the\" An analytical solution to the population balance equation with coalescence and breakage-the special case with constant number of particles\" by DP Patil and JRG Andrews [Chemical Engineering Science 53 (3) 599\u2013601]. Chem. Eng. Sci. 57, 4253\u20134254 (2002)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR27","doi-asserted-by":"publisher","first-page":"3389","DOI":"10.1088\/0305-4470\/14\/12\/030","volume":"14","author":"F Leyvraz","year":"1981","unstructured":"Leyvraz, F., Tschudi, H.R.: Singularities in the kinetics of coagulation processes. J. Phys. A: Math. Gen. 14, 3389 (1981)","journal-title":"J. Phys. A: Math. Gen."},{"key":"3197_CR28","doi-asserted-by":"publisher","first-page":"3715","DOI":"10.1016\/S0009-2509(02)00236-1","volume":"57","author":"YI Lim","year":"2002","unstructured":"Lim, Y.I., Le Lann, J.M., Meyer, X.M., Joulia, X., Lee, G., Yoon, E.S.: On the solution of population balance equations (PBE) with accurate front tracking methods in practical crystallization processes. Chem. Eng. Sci. 57, 3715\u20133732 (2002)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR29","doi-asserted-by":"publisher","first-page":"B448","DOI":"10.1137\/17M1150360","volume":"41","author":"H Liu","year":"2019","unstructured":"Liu, H., Gr\u00f6pler, R., Warnecke, G.: A high-order positivity-preserving DG method for coagulation\u2013fragmentation equations. SIAM J. Sci. Comput. 41, B448\u2013B465 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"3197_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112496","volume":"493","author":"C Liu","year":"2023","unstructured":"Liu, C., Zhang, X.: A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier-Stokes equations. J. Comput. Phys. 493, 112496 (2023)","journal-title":"J. Comput. Phys."},{"key":"3197_CR31","doi-asserted-by":"publisher","first-page":"760","DOI":"10.1137\/0733038","volume":"33","author":"XD Liu","year":"1996","unstructured":"Liu, X.D., Osher, S.: Non-oscillatory high-order accurate self-similar maximum-principle-satisfying shock-capturing schemes I. SIAM J. Numer. Anal. 33, 760\u2013779 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"3197_CR32","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1016\/j.combustflame.2019.04.019","volume":"205","author":"A Liu","year":"2019","unstructured":"Liu, A., Rigopoulos, S.: A conservative method for numerical solution of the population balance equation, and application to soot formation. Combust. Flame 205, 506\u2013521 (2019)","journal-title":"Combust. Flame"},{"key":"3197_CR33","doi-asserted-by":"publisher","first-page":"4298","DOI":"10.1093\/mnras\/staa3682","volume":"501","author":"M Lombart","year":"2021","unstructured":"Lombart, M., Laibe, G.: Grain growth for astrophysics with discontinuous Galerkin schemes. Mon. Not. R. Astron. Soc. 501, 4298\u20134316 (2021)","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"3197_CR34","unstructured":"Lombart, M.: Grain growth by the Galerkin method for the formation of planets, Doctoral dissertation, Universit\u00e9 de Lyon, (2020)"},{"key":"3197_CR35","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1016\/j.ces.2004.02.015","volume":"59","author":"A Maisels","year":"2004","unstructured":"Maisels, A., Kruis, F.E., Fissan, H.: Direct simulation Monte Carlo for simultaneous nucleation, coagulation, and surface growth in dispersed systems. Chem. Eng. Sci. 59, 2231\u20132239 (2004)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR36","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1002\/aic.690490517","volume":"49","author":"DL Marchisio","year":"2003","unstructured":"Marchisio, D.L., Pikturna, J.T., Fox, R.O., Vigil, R.D., Barresi, A.A.: Quadrature method of moments for population-balance equations. AIChE J. 49, 1266\u20131276 (2003)","journal-title":"AIChE J."},{"key":"3197_CR37","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1093\/qmath\/13.1.119","volume":"13","author":"JB McLeod","year":"1962","unstructured":"McLeod, J.B.: On an infinite set of non-linear differential equations. Q. J. Math. 13, 119\u2013128 (1962)","journal-title":"Q. J. Math."},{"key":"3197_CR38","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1090\/S0002-9947-1957-0087880-6","volume":"85","author":"Z Melzak","year":"1957","unstructured":"Melzak, Z.: A scalar transport equation. Trans. Am. Math. Soc. 85, 547\u2013560 (1957)","journal-title":"Trans. Am. Math. Soc."},{"key":"3197_CR39","unstructured":"Myerson, A.: Handbook of industrial crystallization, Butterworth-Heinemann, (2002)"},{"key":"3197_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.ces.2022.117925","volume":"263","author":"D O\u2019Sullivan","year":"2022","unstructured":"O\u2019Sullivan, D., Rigopoulos, S.: A conservative finite volume method for the population balance equation with aggregation, fragmentation, nucleation and growth. Chem. Eng. Sci. 263, 117925 (2022)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR41","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1016\/S0009-2509(97)00314-X","volume":"53","author":"DP Patil","year":"1998","unstructured":"Patil, D.P., Andrews, J.R.G.: An analytical solution to continuous population balance model describing floc coalescence and breakage\u2014a special case. Chem. Eng. Sci. 53, 599\u2013601 (1998)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR42","doi-asserted-by":"publisher","first-page":"7456","DOI":"10.1016\/j.jcp.2011.06.011","volume":"230","author":"RI Patterson","year":"2011","unstructured":"Patterson, R.I., Wagner, W., Kraft, M.: Stochastic weighted particle methods for population balance equations. J. Comput. Phys. 230, 7456\u20137472 (2011)","journal-title":"J. Comput. Phys."},{"key":"3197_CR43","doi-asserted-by":"publisher","first-page":"1576","DOI":"10.1016\/j.compchemeng.2007.01.006","volume":"31","author":"S Qamar","year":"2007","unstructured":"Qamar, S., Warnecke, G.: Numerical solution of population balance equations for nucleation, growth and aggregation processes. Computers & Chemical Engineering 31, 1576\u20131589 (2007)","journal-title":"Computers & Chemical Engineering"},{"key":"3197_CR44","doi-asserted-by":"publisher","first-page":"840","DOI":"10.1002\/aic.690220505","volume":"22","author":"TE Ramabhadran","year":"1976","unstructured":"Ramabhadran, T.E., Peterson, T.W., Seinfeld, J.H.: Dynamics of aerosol coagulation and condensation. AIChE J. 22, 840\u2013851 (1976)","journal-title":"AIChE J."},{"key":"3197_CR45","volume-title":"Triangular mesh methods for the neutron transport equation","author":"WH Reed","year":"1973","unstructured":"Reed, W.H., Hill, T.: Triangular mesh methods for the neutron transport equation. Technical Report, Los Alamos Scientific Laboratory (1973)"},{"key":"3197_CR46","doi-asserted-by":"crossref","unstructured":"Rigopoulos, S.: Population balance of particles in flows: From aerosols to crystallisation, Cambridge University Press, (2024)","DOI":"10.1017\/9781009067829"},{"key":"3197_CR47","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1002\/aic.690490507","volume":"49","author":"S Rigopoulos","year":"2003","unstructured":"Rigopoulos, S., Jones, A.G.: Finite-element scheme for solution of the dynamic population balance equation. AIChE J. 49, 1127\u20131139 (2003)","journal-title":"AIChE J."},{"key":"3197_CR48","doi-asserted-by":"crossref","unstructured":"Rivi\u00e8re, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, SIAM, (2008)","DOI":"10.1137\/1.9780898717440"},{"key":"3197_CR49","doi-asserted-by":"publisher","first-page":"54","DOI":"10.1175\/1520-0469(1968)025<0054:ASOCDC>2.0.CO;2","volume":"25","author":"WT Scott","year":"1968","unstructured":"Scott, W.T.: Analytic studies of cloud droplet coalescence I. J. Atmos. Sci. 25, 54\u201365 (1968)","journal-title":"J. Atmos. Sci."},{"key":"3197_CR50","doi-asserted-by":"publisher","DOI":"10.1016\/j.ces.2023.119020","volume":"280","author":"F Sewerin","year":"2023","unstructured":"Sewerin, F.: An efficient implementation of a conservative finite volume scheme with constant and linear reconstructions for solving the coagulation equation. Chem. Eng. Sci. 280, 119020 (2023)","journal-title":"Chem. Eng. Sci."},{"key":"3197_CR51","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1137\/0909073","volume":"9","author":"C-W Shu","year":"1988","unstructured":"Shu, C.-W.: Total-Variation-Diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073\u20131084 (1988)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"3197_CR52","first-page":"1","volume":"201","author":"C-W Shu","year":"2009","unstructured":"Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability, Numerical Solutions of Partial. Differential Equations 201, 1\u201344 (2009)","journal-title":"Differential Equations"},{"key":"3197_CR53","doi-asserted-by":"publisher","first-page":"439","DOI":"10.1016\/0021-9991(88)90177-5","volume":"77","author":"C-W Shu","year":"1988","unstructured":"Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439\u2013471 (1988)","journal-title":"J. Comput. Phys."},{"key":"3197_CR54","first-page":"129","volume":"92","author":"M von Smoluchowski","year":"1917","unstructured":"von Smoluchowski, M.: Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem. 92, 129\u2013168 (1917)","journal-title":"Z. Phys. Chem."},{"key":"3197_CR55","doi-asserted-by":"publisher","first-page":"1829","DOI":"10.1016\/S0898-1221(03)90004-6","volume":"45","author":"S Wandzurat","year":"2003","unstructured":"Wandzurat, S., Xiao, H.: Symmetric quadrature rules on a triangle. Computers & Mathematics with Applications 45, 1829\u20131840 (2003)","journal-title":"Computers & Mathematics with Applications"},{"key":"3197_CR56","first-page":"1","volume":"7","author":"Y Xu","year":"2010","unstructured":"Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Communications in Computational Physics 7, 1\u201346 (2010)","journal-title":"Communications in Computational Physics"},{"key":"3197_CR57","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111591","volume":"470","author":"Z Xu","year":"2022","unstructured":"Xu, Z., Shu, C.-W.: Third-order maximum-principle-satisfying and positivity-preserving Lax-Wendroff discontinuous Galerkin methods for hyperbolic conservation laws. J. Comput. Phys. 470, 111591 (2022)","journal-title":"J. Comput. Phys."},{"key":"3197_CR58","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112304","volume":"490","author":"Z Xu","year":"2023","unstructured":"Xu, Z., Shu, C.-W.: On the conservation property of positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations. J. Comput. Phys. 490, 112304 (2023)","journal-title":"J. Comput. Phys."},{"key":"3197_CR59","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1007\/s10915-025-03085-8","volume":"105","author":"Z Xu","year":"2025","unstructured":"Xu, Z., Sun, Z., Zhang, Y.-T.: Stability and time-step constraints of exponential time differencing Runge-Kutta discontinuous Galerkin methods for advection-diffusion equations. J. Sci. Comput. 105, 59 (2025)","journal-title":"J. Sci. Comput."},{"key":"3197_CR60","doi-asserted-by":"publisher","first-page":"844","DOI":"10.1016\/j.jcp.2014.10.055","volume":"281","author":"Z Xu","year":"2015","unstructured":"Xu, Z., Zhao, H., Zheng, C.: Accelerating population balance-Monte Carlo simulation for coagulation dynamics from the Markov jump model, stochastic algorithm and GPU parallel computing. J. Comput. Phys. 281, 844\u2013863 (2015)","journal-title":"J. Comput. Phys."},{"key":"3197_CR61","doi-asserted-by":"publisher","first-page":"3091","DOI":"10.1016\/j.jcp.2009.12.030","volume":"229","author":"X Zhang","year":"2010","unstructured":"Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high-order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091\u20133120 (2010)","journal-title":"J. Comput. Phys."},{"key":"3197_CR62","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1016\/j.jcp.2016.10.002","volume":"328","author":"X Zhang","year":"2017","unstructured":"Zhang, X.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations. J. Comput. Phys. 328, 301\u2013343 (2017)","journal-title":"J. Comput. Phys."},{"key":"3197_CR63","doi-asserted-by":"publisher","first-page":"3027","DOI":"10.1088\/0305-4470\/18\/15\/026","volume":"18","author":"RM Ziff","year":"1985","unstructured":"Ziff, R.M., McGrady, E.D.: The kinetics of cluster fragmentation and depolymerisation. J. Phys. A: Math. Gen. 18, 3027 (1985)","journal-title":"J. Phys. A: Math. Gen."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03197-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03197-9","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03197-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T08:03:45Z","timestamp":1772611425000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03197-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,31]]},"references-count":63,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["3197"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03197-9","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,31]]},"assertion":[{"value":"22 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 January 2026","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 declare that we have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"67"}}