{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,23]],"date-time":"2025-08-23T05:21:12Z","timestamp":1755926472706,"version":"3.37.3"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T00:00:00Z","timestamp":1594684800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T00:00:00Z","timestamp":1594684800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11261160486, 91641107, 91852116"],"award-info":[{"award-number":["11261160486, 91641107, 91852116"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Fundamental Research of Civil Aircraft","award":["MJ-F-2012-04"],"award-info":[{"award-number":["MJ-F-2012-04"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s40314-020-01253-0","type":"journal-article","created":{"date-parts":[[2020,7,14]],"date-time":"2020-07-14T21:02:39Z","timestamp":1594760559000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A high-order modified finite-volume method on Cartesian grids for nonlinear convection\u2013diffusion problems"],"prefix":"10.1007","volume":"39","author":[{"given":"Yulong","family":"Du","sequence":"first","affiliation":[]},{"given":"Yahui","family":"Wang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1397-9089","authenticated-orcid":false,"given":"Li","family":"Yuan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,7,14]]},"reference":[{"key":"1253_CR1","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/j.cam.2019.03.017","volume":"358","author":"L Angermann","year":"2019","unstructured":"Angermann L, Wang S (2019) A super-convergent unsymmetric finite volume method for convection-diffusion equations. J Comput Appl Math 358:179\u2013189. https:\/\/doi.org\/10.1016\/j.cam.2019.03.017","journal-title":"J Comput Appl Math"},{"key":"1253_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/C2013-0-19038-1","volume-title":"Computational fluid dynamics: principles and applications","author":"J Blazek","year":"2005","unstructured":"Blazek J (2005) Computational fluid dynamics: principles and applications, 3rd edn. Butterworth Heinemann of Elsevier, Oxford. https:\/\/doi.org\/10.1016\/C2013-0-19038-1","edition":"3"},{"issue":"6","key":"1253_CR3","doi-asserted-by":"publisher","first-page":"3191","DOI":"10.1016\/j.jcp.2007.11.038","volume":"227","author":"R Borges","year":"2008","unstructured":"Borges R, Carmona M, Costa B, Don WS (2008) An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J Comput Phys 227(6):3191\u20133211. https:\/\/doi.org\/10.1016\/j.jcp.2007.11.038","journal-title":"J Comput Phys"},{"key":"1253_CR4","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/s10915-014-9825-1","volume":"61","author":"P Buchm\u00fcller","year":"2014","unstructured":"Buchm\u00fcller P, Helzel C (2014) Improved accuracy of high-order WENO finite volume methods on Cartesian grids. J Sci Comput 61:343\u2013368. https:\/\/doi.org\/10.1007\/s10915-014-9825-1","journal-title":"J Sci Comput"},{"key":"1253_CR5","doi-asserted-by":"publisher","first-page":"460","DOI":"10.1016\/j.amc.2015.03.078","volume":"272","author":"P Buchm\u00fcller","year":"2016","unstructured":"Buchm\u00fcller P, Dreherb J, Helzel C (2016) Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement. Appl Math Comput 272:460\u2013478. https:\/\/doi.org\/10.1016\/j.amc.2015.03.078","journal-title":"Appl Math Comput"},{"key":"1253_CR6","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/978-3-319-91545-6_21","volume":"236","author":"P Buchm\u00fcller","year":"2018","unstructured":"Buchm\u00fcller P, Dreherb J, Helzel C (2018) Improved accuracy of high-order WENO finite volume methods on Cartesian grids with adaptive mesh refinement. Math Stat 236:263\u2013272. https:\/\/doi.org\/10.1007\/978-3-319-91545-6_21","journal-title":"Math Stat"},{"key":"1253_CR7","doi-asserted-by":"publisher","first-page":"1513","DOI":"10.1016\/j.camwa.2014.07.008","volume":"68","author":"J Chana","year":"2014","unstructured":"Chana J, Evans JA, Qiu W (2014) A dual Petrov-Galerkin finite element method for the convection\u2013diffusion equation. Comput Math Appl 68:1513\u20131529. https:\/\/doi.org\/10.1016\/j.camwa.2014.07.008","journal-title":"Comput Math Appl"},{"key":"1253_CR8","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/j.amc.2019.02.043","volume":"354","author":"MS Cheichan","year":"2019","unstructured":"Cheichan MS, Kashkool HA, Gao F (2019) A weak Galerkin finite element method for solving nonlinear convection\u2013diffusion problems in two dimensions. Appl Math Comput 354:149\u2013163. https:\/\/doi.org\/10.1016\/j.amc.2019.02.043","journal-title":"Appl Math Comput"},{"key":"1253_CR9","doi-asserted-by":"publisher","first-page":"992","DOI":"10.1016\/j.jcp.2006.11.006","volume":"224","author":"CS Chou","year":"2007","unstructured":"Chou CS, Shu CW (2007) High order residual distribution conservative finite difference WENO schemes for convection\u2013diffusion steady state problems on non-smooth meshes. J Comput Phys 224:992\u20131020. https:\/\/doi.org\/10.1016\/j.jcp.2006.11.006","journal-title":"J Comput Phys"},{"key":"1253_CR10","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1002\/num.21988","volume":"32","author":"X Cui","year":"2016","unstructured":"Cui X, Yuan GW, Yue JY (2016) Numerical analysis and iteration acceleration of a fully implicit scheme for nonlinear diffusion problem with second-order time evolution. Numer Methods Partial Differ Equ 32:121\u2013140. https:\/\/doi.org\/10.1002\/num.21988","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1253_CR11","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1016\/j.jcp.2013.05.018","volume":"250","author":"WS Don","year":"2013","unstructured":"Don WS, Borges R (2013) Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J Comput Phys 250:347\u2013372. https:\/\/doi.org\/10.1016\/j.jcp.2013.05.018","journal-title":"J Comput Phys"},{"key":"1253_CR12","doi-asserted-by":"publisher","first-page":"768","DOI":"10.4208\/cicp.OA-2018-0254","volume":"26","author":"YL Du","year":"2019","unstructured":"Du YL, Yuan L, Wang YH (2019) A high-order modified finite volume WENO method on 3D Cartesian grids. Commun Comput Phys 26:768\u2013784. https:\/\/doi.org\/10.4208\/cicp.OA-2018-0254","journal-title":"Commun Comput Phys"},{"key":"1253_CR13","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1016\/j.cam.2019.03.018","volume":"359","author":"Y Gao","year":"2019","unstructured":"Gao Y, Liang D, Li Y (2019) Optimal weighted upwind finite volume method for convection\u2013diffusion equations in 2D. J Comput Appl Math 359:73\u201387. https:\/\/doi.org\/10.1016\/j.cam.2019.03.018","journal-title":"J Comput Appl Math"},{"key":"1253_CR14","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1007\/s10915-007-9169-1","volume":"34","author":"G Gassner","year":"2008","unstructured":"Gassner G, L\u00f6rcher F, Munz CD (2008) A discontinuous Galerkin scheme based on a space-time expansion II: viscous flow equations in multi dimensions. J Sci Comput 34:260\u2013286. https:\/\/doi.org\/10.1007\/s10915-007-9169-1","journal-title":"J Sci Comput"},{"key":"1253_CR15","doi-asserted-by":"publisher","first-page":"1224","DOI":"10.1016\/j.cpc.2010.03.008","volume":"181","author":"A Golbabai","year":"2010","unstructured":"Golbabai A, Arabshahi MM (2010) A numerical method for diffusion\u2013convection equation using high-order difference schemes. Comput Phys Commun 181:1224\u20131230. https:\/\/doi.org\/10.1016\/j.cpc.2010.03.008","journal-title":"Comput Phys Commun"},{"key":"1253_CR16","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/j.jcp.2019.02.043","volume":"387","author":"Z Huang","year":"2019","unstructured":"Huang Z, Lin G, Ardekani AM (2019) A mixed upwind\/central WENO scheme for incompressible two-phase flows. J Comput Phys 387:455\u2013480. https:\/\/doi.org\/10.1016\/j.jcp.2019.02.043","journal-title":"J Comput Phys"},{"key":"1253_CR17","doi-asserted-by":"publisher","first-page":"553","DOI":"10.1002\/nme.1641","volume":"67","author":"D Liang","year":"2006","unstructured":"Liang D, Zhao W (2006) An optimal weighted upwind covolume method on non-standard grids for convection\u2013diffusion problems in 2D. Int J Numer Meth Eng 67:553\u2013577. https:\/\/doi.org\/10.1002\/nme.1641","journal-title":"Int J Numer Meth Eng"},{"key":"1253_CR18","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1002\/num.20345","volume":"25","author":"Y Lin","year":"2009","unstructured":"Lin Y, Gao X, Xiao MQ (2009) A high-order finite difference method for 1D nonhomogeneous heat equations. Numer Methods Partial Differ Equ 25:327\u2013346. https:\/\/doi.org\/10.1002\/num.20345","journal-title":"Numer Methods Partial Differ Equ"},{"key":"1253_CR19","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1002\/fld.2021","volume":"62","author":"SC Lo","year":"2010","unstructured":"Lo SC, Blaisdell GA, Lyrintzis AS (2010) High-order shock capturing schemes for turbulence calculations. Int J Numer Meth Fluids 62:473\u2013498. https:\/\/doi.org\/10.1002\/fld.2021","journal-title":"Int J Numer Meth Fluids"},{"key":"1253_CR20","doi-asserted-by":"publisher","first-page":"1242","DOI":"10.1016\/j.cma.2007.11.014","volume":"197","author":"G Manzini","year":"2008","unstructured":"Manzini G, Russo A (2008) A finite volume method for advection-diffusion problems in convection-dominated regimes. Comput Methods Appl Mech Eng 197:1242\u20131261. https:\/\/doi.org\/10.1016\/j.cma.2007.11.014","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1253_CR21","doi-asserted-by":"publisher","DOI":"10.1201\/9780203711194","volume-title":"Numerical solution of convection\u2013diffusion problems","author":"KW Morton","year":"1996","unstructured":"Morton KW (1996) Numerical solution of convection\u2013diffusion problems. Chapman & Hall, London. https:\/\/doi.org\/10.1201\/9780203711194"},{"key":"1253_CR22","doi-asserted-by":"publisher","first-page":"252","DOI":"10.1016\/j.jcp.2019.03.011","volume":"388","author":"K Schmidmayer","year":"2019","unstructured":"Schmidmayer K, Petitpas F, Daniel E (2019) Adaptive mesh refinement algorithm based on dual trees for cells and faces for multiphase compressible flows. J Comput Phys 388:252\u2013278. https:\/\/doi.org\/10.1016\/j.jcp.2019.03.011","journal-title":"J Comput Phys"},{"key":"1253_CR23","unstructured":"Shu CW (1997) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA\/CR-97-206253, ICASE Report NO. 97-65"},{"key":"1253_CR24","doi-asserted-by":"publisher","first-page":"790","DOI":"10.1016\/j.cpc.2013.11.009","volume":"185","author":"HW Sun","year":"2014","unstructured":"Sun HW, Li LZ (2014) A CCD-ADI method for unsteady convection\u2013diffusion equations. Comput Phys Commun 185:790\u2013797. https:\/\/doi.org\/10.1016\/j.cpc.2013.11.009","journal-title":"Comput Phys Commun"},{"key":"1253_CR25","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.jcp.2005.10.019","volume":"215","author":"Y Sun","year":"2006","unstructured":"Sun Y, Wang ZJ, Liu Y (2006) Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow. J Comput Phys 215:41\u201358. https:\/\/doi.org\/10.1016\/j.jcp.2005.10.019","journal-title":"J Comput Phys"},{"key":"1253_CR26","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1016\/j.compfluid.2016.12.002","volume":"144","author":"Y Tamaki","year":"2017","unstructured":"Tamaki Y, Imamura T (2017) Efficient dimension-by-dimension higher order finite-volume methods for a Cartesian grid with cell-based refinement. Comput Fluids 144:74\u201385. https:\/\/doi.org\/10.1016\/j.compfluid.2016.12.002","journal-title":"Comput Fluids"},{"key":"1253_CR27","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s10915-010-9407-9","volume":"46","author":"F Teng","year":"2011","unstructured":"Teng F, Yuan L, Tang T (2011) A speed-up strategy for finite volume WENO schemes for hyperbolic conservation laws. J Sci Comput 46:359\u2013378. https:\/\/doi.org\/10.1007\/s10915-010-9407-9","journal-title":"J Sci Comput"},{"key":"1253_CR28","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1016\/j.chaos.2018.09.011","volume":"118","author":"J Tian","year":"2019","unstructured":"Tian J (2019) An upwind finite volume method for convection\u2013diffusion equations on rectangular mesh. Chaos Solitons Fractals 118:159\u2013165. https:\/\/doi.org\/10.1016\/j.chaos.2018.09.011","journal-title":"Chaos Solitons Fractals"},{"key":"1253_CR29","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.jcp.2004.05.015","volume":"201","author":"VA Titarev","year":"2004","unstructured":"Titarev VA, Toro EF (2004) Finite-volume WENO schemes for three-dimensional conservation laws. J Comput Phys 201:238\u2013260. https:\/\/doi.org\/10.1016\/j.jcp.2004.05.015","journal-title":"J Comput Phys"},{"key":"1253_CR30","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1016\/j.jcp.2018.12.034","volume":"381","author":"US Vevek","year":"2019","unstructured":"Vevek US, Zang B, New TH (2019) Adaptive mapping for high order WENO methods. J Comput Phys 381:162\u2013188. https:\/\/doi.org\/10.1016\/j.jcp.2018.12.034","journal-title":"J Comput Phys"},{"key":"1253_CR31","doi-asserted-by":"publisher","first-page":"2699","DOI":"10.1016\/j.na.2006.09.034","volume":"67","author":"X Wang","year":"2007","unstructured":"Wang X, Li Z (2007) Dynamics for a type of general reaction\u2013diffusion model. Nonlinear Anal 67:2699\u20132711. https:\/\/doi.org\/10.1016\/j.na.2006.09.034","journal-title":"Nonlinear Anal"},{"key":"1253_CR32","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1137\/S1064827598349215","volume":"22","author":"H Wang","year":"2000","unstructured":"Wang H, Liang D, Ewing RE, Lyons SL, Qin G (2000) An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian- Lagrangian localized adjoint method and mixed finite element methods. SIAM J Sci Comput 22:561\u2013581. https:\/\/doi.org\/10.1137\/S1064827598349215","journal-title":"SIAM J Sci Comput"},{"key":"1253_CR33","doi-asserted-by":"publisher","first-page":"3558","DOI":"10.1016\/j.camwa.2018.02.017","volume":"75","author":"J Xie","year":"2018","unstructured":"Xie J, Zhang Z (2018) The high-order multistep ADI solver for two-dimensional nonlinear delayed reaction\u2013diffusion equations with variable. Comput Math Appl 75:3558\u20133570. https:\/\/doi.org\/10.1016\/j.camwa.2018.02.017","journal-title":"Comput Math Appl"},{"key":"1253_CR34","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1006\/jcph.1998.6177","volume":"150","author":"HC Yee","year":"1999","unstructured":"Yee HC, Sandham ND, Djomehri MJ (1999) Low-dissipative high-order shock-capturing methods using characteristic-based filters. J Comput Phys 150:199\u2013238. https:\/\/doi.org\/10.1006\/jcph.1998.6177","journal-title":"J Comput Phys"},{"key":"1253_CR35","doi-asserted-by":"publisher","first-page":"375","DOI":"10.4208\/cicp.OA-2016-0080","volume":"22","author":"H Yue","year":"2017","unstructured":"Yue H, Cheng J, Liu T (2017) A symmetric direct discontinuous Galerkin method for the compressible Navier\u2013Stokes equations. Commun Comput Phys 22:375\u2013392. https:\/\/doi.org\/10.4208\/cicp.OA-2016-0080","journal-title":"Commun Comput Phys"},{"key":"1253_CR36","doi-asserted-by":"publisher","first-page":"612","DOI":"10.1016\/j.amc.2018.10.064","volume":"346","author":"T Zhang","year":"2019","unstructured":"Zhang T, Chen Y (2019) An analysis of the weak Galerkin finite element method for convection\u2013diffusion equations. Appl Math Comput 346:612\u2013621. https:\/\/doi.org\/10.1016\/j.amc.2018.10.064","journal-title":"Appl Math Comput"},{"key":"1253_CR37","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s10915-006-9111-y","volume":"31","author":"S Zhang","year":"2007","unstructured":"Zhang S, Shu CW (2007) A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J Sci Comput 31:273\u2013305. https:\/\/doi.org\/10.1007\/s10915-006-9111-y","journal-title":"J Sci Comput"},{"issue":"3","key":"1253_CR38","doi-asserted-by":"publisher","first-page":"807","DOI":"10.4208\/cicp.291109.080410s","volume":"9","author":"R Zhang","year":"2011","unstructured":"Zhang R, Zhang M, Shu CW (2011) On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun Comput Phys 9(3):807\u2013827. https:\/\/doi.org\/10.4208\/cicp.291109.080410s","journal-title":"Commun Comput Phys"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01253-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-020-01253-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-020-01253-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,13]],"date-time":"2021-07-13T23:29:33Z","timestamp":1626218973000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-020-01253-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,14]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1253"],"URL":"https:\/\/doi.org\/10.1007\/s40314-020-01253-0","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"type":"print","value":"2238-3603"},{"type":"electronic","value":"1807-0302"}],"subject":[],"published":{"date-parts":[[2020,7,14]]},"assertion":[{"value":"8 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 July 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest to this work.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"214"}}