{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:27:06Z","timestamp":1786580826671,"version":"build-2736575974"},"reference-count":40,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.cnsns.2026.110592","type":"journal-article","created":{"date-parts":[[2026,7,20]],"date-time":"2026-07-20T15:06:05Z","timestamp":1784559965000},"page":"110592","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P2","title":["Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent"],"prefix":"10.1016","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-2012-2244","authenticated-orcid":false,"given":"Hao","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-5344-3636","authenticated-orcid":false,"given":"Kexin","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1144-5315","authenticated-orcid":false,"given":"Wenlin","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.110592_bib0001","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1016586905654","article-title":"Variable order and distributed order fractional operators","volume":"29","author":"Lorenzo","year":"2002","journal-title":"Nonlinear Dyn"},{"key":"10.1016\/j.cnsns.2026.110592_bib0002","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1103\/PhysRevLett.51.605","article-title":"Fractional quantization of the hall effect: a hierarchy of incompressible quantum fluid states","volume":"51","author":"Haldane","year":"1983","journal-title":"Phys Rev Lett"},{"key":"10.1016\/j.cnsns.2026.110592_bib0003","series-title":"Fractional Calculus: An Introduction for Physicists","author":"Herrmann","year":"2011"},{"key":"10.1016\/j.cnsns.2026.110592_bib0004","series-title":"Fractional Calculus in Bioengineering","author":"Magin","year":"2006"},{"key":"10.1016\/j.cnsns.2026.110592_bib0005","series-title":"Lectures on Viscoelasticity Theory","author":"Pipkin","year":"1986"},{"key":"10.1016\/j.cnsns.2026.110592_bib0006","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/S0370-1573(02)00331-9","article-title":"Chaos, fractional kinetics, and anomalous transport","volume":"371","author":"Zaslavsky","year":"2002","journal-title":"Phys Rep"},{"key":"10.1016\/j.cnsns.2026.110592_bib0007","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.jconhyd.2013.11.002","article-title":"Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media","volume":"157","author":"Sun","year":"2014","journal-title":"J Contam Hydrol"},{"key":"10.1016\/j.cnsns.2026.110592_bib0008","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1515\/fca-2019-0003","article-title":"A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications","volume":"22","author":"Sun","year":"2019","journal-title":"Fract Calc Appl Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0009","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1007\/s42102-022-00085-2","article-title":"Fractional modeling in action: a survey of nonlocal models for subsurface transport, turbulent flows, and anomalous materials","volume":"5","author":"Suzuki","year":"2023","journal-title":"J Peridyn Nonlocal Model"},{"key":"10.1016\/j.cnsns.2026.110592_bib0010","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1137\/24M1673371","article-title":"Identification of a spatially-dependent variable order in one-dimensional subdiffusion","volume":"57","author":"Hong","year":"2025","journal-title":"SIAM J Math Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0011","doi-asserted-by":"crossref","first-page":"2211","DOI":"10.1051\/m2an\/2021045","article-title":"Numerical discretization and fast approximation of a variably distributedorder fractional wave equation","volume":"55","author":"Jia","year":"2021","journal-title":"Esaim: M2an"},{"key":"10.1016\/j.cnsns.2026.110592_bib0012","doi-asserted-by":"crossref","first-page":"2725","DOI":"10.1093\/imanum\/drad072","article-title":"A general collocation analysis for weakly singular volterra integral equations with variable exponent","volume":"44","author":"Liang","year":"2024","journal-title":"IMA J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0013","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1007\/s10915-024-02593-3","article-title":"Sharp error bounds for a fractional collocation method for weakly singular volterra integral equations with variable exponent","volume":"100","author":"Ma","year":"2024","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0014","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/j.jcp.2014.12.001","article-title":"Fractional spectral collocation methods for linear and nonlinear variable order fpdes","volume":"293","author":"Zayernouri","year":"2015","journal-title":"J Comput Phys"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110592_bib0015","doi-asserted-by":"crossref","first-page":"A2710","DOI":"10.1137\/141001299","article-title":"A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations","volume":"37","author":"Zeng","year":"2015","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0016","doi-asserted-by":"crossref","first-page":"1534","DOI":"10.1137\/24M1635612","article-title":"Local modification of subdiffusion by initial fickian diffusion: multiscale modeling, analysis, and computation","volume":"22","author":"Zheng","year":"2024","journal-title":"Multiscale Model Simul"},{"key":"10.1016\/j.cnsns.2026.110592_bib0017","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1137\/080730597","article-title":"Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term","volume":"47","author":"Zhuang","year":"2009","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0018","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/j.jcp.2014.08.015","article-title":"Second-order approximations for variable order fractional derivatives: algorithms and applications","volume":"293","author":"Zhao","year":"2015","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110592_bib0019","doi-asserted-by":"crossref","DOI":"10.1098\/rspa.2019.0498","article-title":"Applications of variable-order fractional operators: a review","volume":"476","author":"Patnaik","year":"2020","journal-title":"Proc R Soc A"},{"key":"10.1016\/j.cnsns.2026.110592_bib0020","doi-asserted-by":"crossref","first-page":"1740","DOI":"10.1137\/090771715","article-title":"Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation","volume":"32","author":"Chen","year":"2010","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0021","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1051\/m2an\/2020072","article-title":"Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order","volume":"55","author":"Li","year":"2021","journal-title":"Esaim: M2an"},{"key":"10.1016\/j.cnsns.2026.110592_bib0022","doi-asserted-by":"crossref","first-page":"1778","DOI":"10.1016\/j.jmaa.2019.03.052","article-title":"Wellposedness and regularity of the variable-order time-fractional diffusion equations","volume":"475","author":"Wang","year":"2019","journal-title":"J Math Anal Appl"},{"key":"10.1016\/j.cnsns.2026.110592_bib0023","doi-asserted-by":"crossref","first-page":"2492","DOI":"10.1137\/20M132420X","article-title":"An error estimate of a numerical approximation to a hidden-memory variable-order space-time fractional diffusion equation","volume":"58","author":"Zheng","year":"2020","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0024","doi-asserted-by":"crossref","first-page":"1522","DOI":"10.1093\/imanum\/draa013","article-title":"Optimal-order error estimates of finite element approximations to variable-order time-fractional diffusion equations without regularity assumptions of the true solutions","volume":"41","author":"Zheng","year":"2021","journal-title":"IMA J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0025","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1137\/19M1301230","article-title":"Temporal second-order finite difference schemes for variable-order time-fractional wave equations","volume":"60","author":"Du","year":"2022","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2021.106047","article-title":"Analysis and discretization of a variable-order fractional wave equation","volume":"104","author":"Zheng","year":"2022","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"10.1016\/j.cnsns.2026.110592_bib0027","doi-asserted-by":"crossref","first-page":"666","DOI":"10.4208\/csiam-am.SO-2024-0052","article-title":"Two methods addressing variable-exponent fractional initial and boundary value problems and abel integral equation","volume":"6","author":"Zheng","year":"2025","journal-title":"CSIAM T Appl Math"},{"key":"10.1016\/j.cnsns.2026.110592_bib0028","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10915-025-03080-z","article-title":"Numerical approximation for variable-exponent fractional diffusion-wave equation","volume":"105","author":"Qiu","year":"2025","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0029","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1007\/s00211-006-0045-y","article-title":"A second-order accurate numerical method for a fractional wave equation","volume":"105","author":"McLean","year":"2007","journal-title":"Numer Math"},{"key":"10.1016\/j.cnsns.2026.110592_bib0030","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1137\/0915016","article-title":"A novel two-grid method for semilinear elliptic equations","volume":"15","author":"Xu","year":"1994","journal-title":"SIAM J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0031","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1137\/S0036142992232949","article-title":"Two-grid discretization techniques for linear and nonlinear PDEs","volume":"33","author":"Xu","year":"1996","journal-title":"SIAM J Numer Anal"},{"key":"10.1016\/j.cnsns.2026.110592_bib0032","series-title":"Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations","volume":"vol. 45","author":"Chen","year":"2019"},{"key":"10.1016\/j.cnsns.2026.110592_bib0033","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s10092-023-00508-6","article-title":"A two-grid temporal second-order scheme for the two-dimensional nonlinear volterra integro-differential equation with weakly singular kernel","volume":"60","author":"Chen","year":"2023","journal-title":"Calcolo"},{"key":"10.1016\/j.cnsns.2026.110592_bib0034","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1016\/j.cam.2013.03.006","article-title":"Long-time behavior of the two-grid finite element method for fully discrete semilinear evolution equations with positive memory","volume":"250","author":"Wang","year":"2013","journal-title":"J Comput Appl Math"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110592_bib0035","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1007\/s10915-023-02282-7","article-title":"A high-order two-grid difference method for nonlinear time-fractional biharmonic problems and its unconditional \u03b1-robust error estimates","volume":"96","author":"Fu","year":"2023","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0036","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s10915-022-02000-9","article-title":"A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations","volume":"93","author":"Lyu","year":"2022","journal-title":"J Sci Comput"},{"key":"10.1016\/j.cnsns.2026.110592_bib0037","series-title":"Numerical methods of partial differential equations","author":"Sun","year":"2012"},{"key":"10.1016\/j.cnsns.2026.110592_bib0038","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/j.apnum.2021.11.003","article-title":"A crank-nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations","volume":"172","author":"Wang","year":"2022","journal-title":"Appl Numer Math"},{"key":"10.1016\/j.cnsns.2026.110592_bib0039","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.apnum.2025.12.003","article-title":"Two adi compact difference methods for variable exponent diffusion wave equations","volume":"222","author":"Zhang","year":"2026","journal-title":"Appl Numer Math"},{"key":"10.1016\/j.cnsns.2026.110592_bib0040","series-title":"Robust computational techniques for boundary layers","author":"Farrell","year":"2000"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009469?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1007570426009469?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T23:29:10Z","timestamp":1786577350000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1007570426009469"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":40,"alternative-id":["S1007570426009469"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110592","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Spatial two-grid compact difference scheme for two-dimensional nonlinear diffusion-wave equations with variable exponent","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110592","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110592"}}