{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,26]],"date-time":"2025-11-26T16:49:35Z","timestamp":1764175775715,"version":"build-2065373602"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T00:00:00Z","timestamp":1733270400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T00:00:00Z","timestamp":1733270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001501","name":"University Grants Commission","doi-asserted-by":"publisher","award":["1087\/(CSIR-UGC NET DEC. 2016), 1058\/(CSIR-UGC NET DEC. 2018)"],"award-info":[{"award-number":["1087\/(CSIR-UGC NET DEC. 2016), 1058\/(CSIR-UGC NET DEC. 2018)"]}],"id":[{"id":"10.13039\/501100001501","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001843","name":"Science and Engineering Research Board","doi-asserted-by":"publisher","award":["CRG\/2018\/004608"],"award-info":[{"award-number":["CRG\/2018\/004608"]}],"id":[{"id":"10.13039\/501100001843","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2025,11]]},"DOI":"10.1007\/s11075-024-01987-6","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T05:25:48Z","timestamp":1733289948000},"page":"1239-1287","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["High accuracy two-level compact implicit method in exponential form for 2D fourth order quasi-linear parabolic equations"],"prefix":"10.1007","volume":"100","author":[{"given":"Divya","family":"Sharma","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kajal","family":"Mittal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Deepti","family":"Kaur","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rajendra K.","family":"Ray","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R. K.","family":"Mohanty","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,12,4]]},"reference":[{"key":"1987_CR1","doi-asserted-by":"publisher","first-page":"112914","DOI":"10.1016\/j.cma.2020.112914","volume":"364","author":"M Abbaszadeh","year":"2020","unstructured":"Abbaszadeh, M., Dehghan, M.: Reduced order modeling of time-dependent incompressible Navier-Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD\/DEIM method. Comput. Methods Appl. Mech. Eng. 364, 112914 (2020). https:\/\/doi.org\/10.1016\/j.cma.2020.112914","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"1987_CR2","doi-asserted-by":"publisher","first-page":"814","DOI":"10.1002\/num.22124","volume":"33","author":"M Aggul","year":"2017","unstructured":"Aggul, M., Labovsky, A.: A high accuracy minimally invasive regularization technique for Navier-Stokes equations at high Reynolds number. Numer. Methods Partial Differ. Equations 33(3), 814\u2013839 (2017). https:\/\/doi.org\/10.1002\/num.22124","journal-title":"Numer. Methods Partial Differ. Equations"},{"key":"1987_CR3","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0001-8708(78)90130-5","volume":"30","author":"DG Aronson","year":"1978","unstructured":"Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33\u201376 (1978). https:\/\/doi.org\/10.1016\/0001-8708(78)90130-5","journal-title":"Adv. Math."},{"key":"1987_CR4","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s11075-013-9694-y","volume":"65","author":"ZZ Bai","year":"2014","unstructured":"Bai, Z.Z., Ran, Y.H., Yuan, L.: On approximated ILU and UGS preconditioning methods for linearized discretized steady incompressible Navier-Stokes equations. Numer. Algorithms 65, 43\u201368 (2014). https:\/\/doi.org\/10.1007\/s11075-013-9694-y","journal-title":"Numer. Algorithms"},{"key":"1987_CR5","doi-asserted-by":"publisher","unstructured":"Balzano, A.: MOSQUITO: An efficient finite difference scheme for numerical simulation of 2D advection. Int. J. Numer. Methods Fluids 31(2), 481\u2013496 (1999). https:\/\/doi.org\/10.1002\/(SICI)1097-0363(19990930)31:2<481::AID-FLD886>3.0.CO;2-I","DOI":"10.1002\/(SICI)1097-0363(19990930)31:2<481::AID-FLD886>3.0.CO;2-I"},{"key":"1987_CR6","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1090\/S0025-5718-1968-0242392-2","volume":"22","author":"AJ Chorin","year":"1968","unstructured":"Chorin, A.J.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745\u2013762 (1968). https:\/\/doi.org\/10.1090\/S0025-5718-1968-0242392-2","journal-title":"Math. Comp."},{"issue":"2","key":"1987_CR7","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1002\/num.1037","volume":"18","author":"W Dai","year":"2002","unstructured":"Dai, W., Nassar, R.: Compact ADI method for solving parabolic differential equations. Numer. Methods Partial Differ. Equations 18(2), 129\u2013142 (2002). https:\/\/doi.org\/10.1002\/num.1037","journal-title":"Numer. Methods Partial Differ. Equations"},{"key":"1987_CR8","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.cam.2004.04.002","volume":"174","author":"P Danumjaya","year":"2005","unstructured":"Danumjaya, P., Pani, A.K.: Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation. J. Comput. Appl. Math. 174, 101\u2013117 (2005). https:\/\/doi.org\/10.1016\/j.cam.2004.04.002","journal-title":"J. Comput. Appl. Math."},{"key":"1987_CR9","doi-asserted-by":"publisher","first-page":"2641","DOI":"10.1103\/PhysRevLett.60.2641","volume":"60","author":"GT Dee","year":"1988","unstructured":"Dee, G.T., Saarloos, W.V.: Bistable systems with propagating fronts leading to pattern formation. Phys. Rev. Lett. 60, 2641\u20132644 (1988). https:\/\/doi.org\/10.1103\/PhysRevLett.60.2641","journal-title":"Phys. Rev. Lett."},{"key":"1987_CR10","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1016\/J.ENGANABOUND.2019.06.007","volume":"107","author":"M Dehghan","year":"2019","unstructured":"Dehghan, M., Mohammadi, V.: Two-dimensional simulation of the damped Kuramoto-Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization. Eng. Anal. Bound. Elem. 107, 168\u2013184 (2019). https:\/\/doi.org\/10.1016\/J.ENGANABOUND.2019.06.007","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1987_CR11","doi-asserted-by":"publisher","first-page":"1159","DOI":"10.1007\/s00366-019-00877-z","volume":"37","author":"M Dehghan","year":"2021","unstructured":"Dehghan, M., Shafieeabyaneh, N.: Local radial basis function-finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher-Kolmogorov equations. Eng. Comput. 37, 1159\u20131179 (2021). https:\/\/doi.org\/10.1007\/s00366-019-00877-z","journal-title":"Eng. Comput."},{"issue":"2","key":"1987_CR12","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1016\/j.amc.2005.02.029","volume":"172","author":"H Eren","year":"2006","unstructured":"Eren, H.: Numerical study of unsteady interacting compressible jet flows using finite difference method for small Reynolds numbers. Appl. Math. Comput. 172(2), 876\u2013891 (2006). https:\/\/doi.org\/10.1016\/j.amc.2005.02.029","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"1987_CR13","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1002\/fld.1061","volume":"50","author":"E Erturk","year":"2006","unstructured":"Erturk, E., G\u00f6k\u00e7\u00f6l, C.: Fourth-order compact formulation of Navier-Stokes equations and driven cavity flow at higher Reynolds numbers. Int. J. Numer. Methods Fluids 50(4), 421\u2013436 (2006). https:\/\/doi.org\/10.1002\/fld.1061","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"2","key":"1987_CR14","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1137\/0704016","volume":"4","author":"G Fairweather","year":"1967","unstructured":"Fairweather, G., Mitchell, A.R.: A new computational procedure for A.D.I. methods. SIAM J. Numer. Anal. 4(2), 163\u2013170 (1967). https:\/\/doi.org\/10.1137\/0704016","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"1987_CR15","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1002\/fld.1650030302","volume":"3","author":"CAJ Fletcher","year":"1983","unstructured":"Fletcher, C.A.J.: Generating exact solutions of the two-dimensional Burgers\u2019 equations. Int. J. Numer. Methods Fluids 3(3), 213\u2013216 (1983). https:\/\/doi.org\/10.1002\/fld.1650030302","journal-title":"Int. J. Numer. Methods Fluids"},{"issue":"3","key":"1987_CR16","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/0021-9991(82)90058-4","volume":"48","author":"U Ghia","year":"1982","unstructured":"Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387\u2013411 (1982). https:\/\/doi.org\/10.1016\/0021-9991(82)90058-4","journal-title":"J. Comput. Phys."},{"issue":"10","key":"1987_CR17","doi-asserted-by":"publisher","first-page":"1944","DOI":"10.1016\/j.camwa.2016.03.004","volume":"71","author":"G Guo","year":"2016","unstructured":"Guo, G., L\u00fc, S.: Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional fourth-order diffusion equation. Comput. Math. Appl. 71(10), 1944\u20131959 (2016). https:\/\/doi.org\/10.1016\/j.camwa.2016.03.004","journal-title":"Comput. Math. Appl."},{"issue":"12","key":"1987_CR18","doi-asserted-by":"publisher","first-page":"2594","DOI":"10.1016\/j.camwa.2016.04.026","volume":"71","author":"D He","year":"2016","unstructured":"He, D.: On the $$L^{\\infty }$$-norm convergence of a three-level linearly implicit finite difference method for the extended Fisher-Kolmogorov equation in both 1D and 2D. Comput. Math. Appl. 71(12), 2594\u20132607 (2016). https:\/\/doi.org\/10.1016\/j.camwa.2016.04.026","journal-title":"Comput. Math. Appl."},{"key":"1987_CR19","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/0021-9991(75)90118-7","volume":"19","author":"RS Hirsh","year":"1975","unstructured":"Hirsh, R.S.: Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90\u2013109 (1975). https:\/\/doi.org\/10.1016\/0021-9991(75)90118-7","journal-title":"J. Comput. Phys."},{"key":"1987_CR20","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1007\/s00366-017-0530-1","volume":"34","author":"M Ilati","year":"2018","unstructured":"Ilati, M., Dehghan, M.: Direct local boundary integral equation method for numerical solution of extended Fisher-Kolmogorov equation. Eng. Comput. 34, 203\u2013213 (2018). https:\/\/doi.org\/10.1007\/s00366-017-0530-1","journal-title":"Eng. Comput."},{"key":"1987_CR21","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1006\/jcph.2002.7079","volume":"180","author":"H Johnston","year":"2002","unstructured":"Johnston, H., Liu, J.G.: Finite difference schemes for incompressible flow based on local pressure boundary conditions. J. Comput. Phys. 180, 120\u2013154 (2002). https:\/\/doi.org\/10.1006\/jcph.2002.7079","journal-title":"J. Comput. Phys."},{"issue":"12","key":"1987_CR22","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1002\/fld.263","volume":"38","author":"JC Kalita","year":"2002","unstructured":"Kalita, J.C., Dalal, D.C., Dass, A.K.: A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients. Int. J. Numer. Methods Fluids 38(12), 1111\u20131131 (2002). https:\/\/doi.org\/10.1002\/fld.263","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1987_CR23","doi-asserted-by":"publisher","first-page":"20140932","DOI":"10.1098\/rspa.2014.0932","volume":"471","author":"A Kalogirou","year":"2015","unstructured":"Kalogirou, A., Keaveny, E.E., Papageorgiou, D.T.: An in-depth numerical study of the two-dimensional Kuramoto-Sivashinsky equation. Proc. R. Soc. A 471, 20140932 (2015). https:\/\/doi.org\/10.1098\/rspa.2014.0932","journal-title":"Proc. R. Soc. A"},{"key":"1987_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2004.01.002","volume":"198","author":"S Karaa","year":"2004","unstructured":"Karaa, S., Zhang, J.: High order ADI method for solving unsteady convection-diffusion problems. J. Comput. Phys. 198, 1\u20139 (2004). https:\/\/doi.org\/10.1016\/j.jcp.2004.01.002","journal-title":"J. Comput. Phys."},{"key":"1987_CR25","doi-asserted-by":"publisher","unstructured":"Kelly, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM Publications, Philadelphia (1995). https:\/\/doi.org\/10.1137\/1.9781611970944","DOI":"10.1137\/1.9781611970944"},{"issue":"11","key":"1987_CR26","doi-asserted-by":"publisher","first-page":"4151","DOI":"10.1016\/j.camwa.2011.09.065","volume":"62","author":"N Khiari","year":"2011","unstructured":"Khiari, N., Omrani, K.: Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions. Comput. Math. Appl. 62(11), 4151\u20134160 (2011). https:\/\/doi.org\/10.1016\/j.camwa.2011.09.065","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1987_CR27","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1143\/PTP.55.356","volume":"55","author":"Y Kuramoto","year":"1976","unstructured":"Kuramoto, Y., Tsuzuki, T.: Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55(2), 356\u2013369 (1976). https:\/\/doi.org\/10.1143\/PTP.55.356","journal-title":"Prog. Theor. Phys."},{"issue":"10","key":"1987_CR28","doi-asserted-by":"publisher","first-page":"1137","DOI":"10.1002\/FLD.1650201003","volume":"20","author":"M Li","year":"1995","unstructured":"Li, M., Tang, T., Fornberg, B.: A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 20(10), 1137\u20131151 (1995). https:\/\/doi.org\/10.1002\/FLD.1650201003","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"1987_CR29","doi-asserted-by":"publisher","first-page":"94","DOI":"10.1186\/s13662-017-1154-x","volume":"2017","author":"F Liu","year":"2017","unstructured":"Liu, F., Zhao, X., Liu, B.: Fourier pseudo-spectral method for the extended Fisher-Kolmogorov equation in two dimensions. Adv. Differ. Equations 2017, 94 (2017). https:\/\/doi.org\/10.1186\/s13662-017-1154-x","journal-title":"Adv. Differ. Equations"},{"key":"1987_CR30","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1006\/jcph.1999.6387","volume":"157","author":"HL Meitz","year":"2000","unstructured":"Meitz, H.L., Fasel, H.F.: A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation. J. Comput. Phys. 157, 371\u2013403 (2000). https:\/\/doi.org\/10.1006\/jcph.1999.6387","journal-title":"J. Comput. Phys."},{"issue":"9","key":"1987_CR31","doi-asserted-by":"publisher","first-page":"4281","DOI":"10.1108\/hff-07-2019-0597","volume":"30","author":"RC Mittal","year":"2020","unstructured":"Mittal, R.C., Kumar, S., Jiwari, R.: A cubic B-spline quasi-interpolation method for solving two-dimensional unsteady advection diffusion equations. Int. J. Numer. Methods Heat Fluid Flow 30(9), 4281\u20134306 (2020). https:\/\/doi.org\/10.1108\/hff-07-2019-0597","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"1987_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/00207160.2015.1085976","volume":"94","author":"RC Mittal","year":"2017","unstructured":"Mittal, R.C., Tripathi, A.: Numerical solutions of two-dimensional unsteady convection-diffusion problems using modified bi-cubic B-spline finite elements. Int. J. Comput. Math. 94, 1\u201321 (2017). https:\/\/doi.org\/10.1080\/00207160.2015.1085976","journal-title":"Int. J. Comput. Math."},{"key":"1987_CR33","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/0377-0427(95)00135-2","volume":"70","author":"RK Mohanty","year":"1996","unstructured":"Mohanty, R.K., Jain, M.K.: High accuracy difference schemes for the system of two space nonlinear parabolic differential equations with mixed derivatives and variable coefficients. J. Comput. Appl. Math. 70, 15\u201332 (1996). https:\/\/doi.org\/10.1016\/0377-0427(95)00135-2","journal-title":"J. Comput. Appl. Math."},{"key":"1987_CR34","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1515\/cmam-2016-0047","volume":"17","author":"RK Mohanty","year":"2017","unstructured":"Mohanty, R.K., Kaur, D.: High accuracy compact operator methods for two-dimensional fourth order nonlinear parabolic partial differential equations. Comput. Methods Appl. Math. 17, 617\u2013641 (2017). https:\/\/doi.org\/10.1515\/cmam-2016-0047","journal-title":"Comput. Methods Appl. Math."},{"issue":"5","key":"1987_CR35","doi-asserted-by":"publisher","first-page":"2680","DOI":"10.1016\/j.amc.2012.08.100","volume":"219","author":"RK Mohanty","year":"2012","unstructured":"Mohanty, R.K., Setia, N.: A new high accuracy two-level implicit off-step discretization for the system of two space dimensional quasi-linear parabolic partial differential equations. Appl. Math. Comput. 219(5), 2680\u20132697 (2012). https:\/\/doi.org\/10.1016\/j.amc.2012.08.100","journal-title":"Appl. Math. Comput."},{"issue":"2","key":"1987_CR36","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1016\/j.cam.2006.10.023","volume":"208","author":"RK Mohanty","year":"2007","unstructured":"Mohanty, R.K., Singh, S.: A new two-level implicit discretization of $$O(k^2+kh^2+h^4)$$ for the solution of singularly perturbed two-space dimensional non-linear parabolic equations. J. Comput. Appl. Math. 208(2), 391\u2013403 (2007). https:\/\/doi.org\/10.1016\/j.cam.2006.10.023","journal-title":"J. Comput. Appl. Math."},{"key":"1987_CR37","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.amc.2019.01.050","volume":"352","author":"RK Mohanty","year":"2019","unstructured":"Mohanty, R.K., Kaur, D., Singh, S.: A class of two- and three-level implicit methods of order two in time and four in space based on half-step discretization for two-dimensional fourth order quasi-linear parabolic equations. Appl. Math. Comput. 352, 68\u201387 (2019). https:\/\/doi.org\/10.1016\/j.amc.2019.01.050","journal-title":"Appl. Math. Comput."},{"key":"1987_CR38","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1186\/S13662-019-1968-9","volume":"2019","author":"RK Mohanty","year":"2019","unstructured":"Mohanty, R.K., Manchanda, G., Khan, A.: Operator compact exponential approximation for the solution of the system of 2D second order quasi-linear elliptic partial differential equations. Adv. Differ. Equations 2019, 47 (2019). https:\/\/doi.org\/10.1186\/S13662-019-1968-9","journal-title":"Adv. Differ. Equations"},{"issue":"3","key":"1987_CR39","doi-asserted-by":"publisher","first-page":"1938","DOI":"10.1002\/num.22795","volume":"39","author":"BN Saray","year":"2021","unstructured":"Saray, B.N., Lakestani, M., Dehghan, M.: On the sparse multiscale representation of 2-D Burgers equations by an efficient algorithm based on multiwavelets. Numer. Methods Partial Differ. Equations 39(3), 1938\u20131961 (2021). https:\/\/doi.org\/10.1002\/num.22795","journal-title":"Numer. Methods Partial Differ. Equations"},{"key":"1987_CR40","doi-asserted-by":"publisher","first-page":"949","DOI":"10.1016\/j.amc.2006.11.169","volume":"189","author":"K Omrani","year":"2007","unstructured":"Omrani, K.: Numerical methods and error analysis for the nonlinear Sivashinsky equation. Appl. Math. Comput. 189, 949\u2013962 (2007). https:\/\/doi.org\/10.1016\/j.amc.2006.11.169","journal-title":"Appl. Math. Comput."},{"key":"1987_CR41","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1137\/0103003","volume":"3","author":"DW Peaceman","year":"1955","unstructured":"Peaceman, D.W., Rachford, H.H., Jr.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28\u201341 (1955). https:\/\/doi.org\/10.1137\/0103003","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"1987_CR42","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1006\/JCPH.2000.6673","volume":"167","author":"JMC Pereira","year":"2001","unstructured":"Pereira, J.M.C., Kobayashi, M.H., Pereira, J.C.F.: A fourth-order-accurate finite volume compact method for the incompressible Navier-Stokes solutions. J. Comput. Phys. 167, 217\u2013243 (2001). https:\/\/doi.org\/10.1006\/JCPH.2000.6673","journal-title":"J. Comput. Phys."},{"issue":"11","key":"1987_CR43","doi-asserted-by":"publisher","first-page":"2620","DOI":"10.1108\/HFF-12-2017-0511","volume":"28","author":"R Rohila","year":"2018","unstructured":"Rohila, R., Mittal, R.C.: An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations. Int. J. Numer. Methods Heat Fluid Flow 28(11), 2620\u20132649 (2018). https:\/\/doi.org\/10.1108\/HFF-12-2017-0511","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"1987_CR44","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1146\/ANNUREV.FL.15.010183.001143","volume":"15","author":"GI Sivashinsky","year":"1983","unstructured":"Sivashinsky, G.I.: Instabilities, pattern-formation, and turbulence in flames. Ann. Rev. Fluid Mech. 15, 179\u2013199 (1983). https:\/\/doi.org\/10.1146\/ANNUREV.FL.15.010183.001143","journal-title":"Ann. Rev. Fluid Mech."},{"issue":"20","key":"1987_CR45","doi-asserted-by":"publisher","first-page":"3497","DOI":"10.1002\/NME.1620382008","volume":"38","author":"WF Spotz","year":"1995","unstructured":"Spotz, W.F., Carey, G.F.: High-order compact scheme for the steady stream-function vorticity equations. Int. J. Numer. Methods Eng. 38(20), 3497\u20133512 (1995). https:\/\/doi.org\/10.1002\/NME.1620382008","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1987_CR46","doi-asserted-by":"publisher","unstructured":"Strikwerda, J.C.: High-order-accurate schemes for incompressible viscous flow. Int. J. Numer. Methods Fluids 24(7), 715\u2013734 (1997). https:\/\/doi.org\/10.1002\/(SICI)1097-0363(19970415)24:7<715::AID-FLD513>3.0.CO;2-E","DOI":"10.1002\/(SICI)1097-0363(19970415)24:7<715::AID-FLD513>3.0.CO;2-E"},{"key":"1987_CR47","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1006\/JCPH.1995.1012","volume":"116","author":"D Tafti","year":"1995","unstructured":"Tafti, D.: Alternate formulations for the pressure equation Laplacian on a collocated grid for solving the unsteady incompressible Navier-Stokes equations. J. Comput. Phys. 116, 143\u2013153 (1995). https:\/\/doi.org\/10.1006\/JCPH.1995.1012","journal-title":"J. Comput. Phys."},{"key":"1987_CR48","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1006\/jcph.1996.0125","volume":"126","author":"E Weinan","year":"1996","unstructured":"Weinan, E., Liu, J.G.: Essentially compact schemes for unsteady viscous incompressible flows. J. Comput. Phys. 126, 122\u2013138 (1996). https:\/\/doi.org\/10.1006\/jcph.1996.0125","journal-title":"J. Comput. Phys."},{"key":"1987_CR49","doi-asserted-by":"publisher","unstructured":"Yanwen, M., Dexun, F., Kobayashi, T., Taniguchi, N.: Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme. Int. J. Numer. Methods Fluids 30(5), 509\u2013521 (1999). https:\/\/doi.org\/10.1002\/(SICI)1097-0363(19990715)30:5<509::AID-FLD851>3.0.CO;2-E","DOI":"10.1002\/(SICI)1097-0363(19990715)30:5<509::AID-FLD851>3.0.CO;2-E"},{"issue":"18","key":"1987_CR50","doi-asserted-by":"publisher","first-page":"1332","DOI":"10.1103\/PHYSREVLETT.49.1332","volume":"49","author":"Z Guozhen","year":"1982","unstructured":"Guozhen, Z.: Experiments on director waves in nematic liquid crystals. Phys. Rev. Lett. 49(18), 1332\u20131335 (1982). https:\/\/doi.org\/10.1103\/PHYSREVLETT.49.1332","journal-title":"Phys. Rev. Lett."},{"key":"1987_CR51","unstructured":"Ba\u015fhan, A., U\u00e7ar, Y., Ya\u011fmurlu, N.M., Esen, A.: Numerical solutions for the fourth order Extended Fisher-Kolmogorov equation with high accuracy by differential quadrature method. Sigma J. Eng. Nat. Sci. 9(3), 273\u2013284 (2018). https:\/\/www.researchgate.net\/publication\/329529153"},{"key":"1987_CR52","doi-asserted-by":"publisher","first-page":"123755","DOI":"10.1016\/j.physa.2019.123755","volume":"545","author":"A Ba\u015fhan","year":"2020","unstructured":"Ba\u015fhan, A.: A numerical treatment of the coupled viscous Burgers\u2019 equation in the presence of very large Reynolds number. Phys. A 545, 123755 (2020). https:\/\/doi.org\/10.1016\/j.physa.2019.123755","journal-title":"Phys. A"},{"key":"1987_CR53","unstructured":"U\u00e7ar, Y., Ya\u011fmurlu, N.M., Ba\u015fhan, A.: Numerical solutions and stability analysis of modified Burgers equation via modified cubic b-spline differential quadrature methods. Sigma J. Eng. Nat. Sci. 37, 129\u2013142 (2019). https:\/\/www.researchgate.net\/publication\/332031463"},{"issue":"7","key":"1987_CR54","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1002\/cnm.850","volume":"22","author":"WL Wood","year":"2006","unstructured":"Wood, W.L.: An exact solution for Burgers\u2019 equation. Commun. Numer. Methods Eng. 22(7), 797\u2013798 (2006). https:\/\/doi.org\/10.1002\/cnm.850","journal-title":"Commun. Numer. Methods Eng."},{"issue":"2","key":"1987_CR55","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1002\/num.22306","volume":"35","author":"LJT Doss","year":"2019","unstructured":"Doss, L.J.T., Nandini, A.P.: A fourth-order $$H^{1}$$-Galerkin mixed finite element method for Kuramoto-Sivashinsky equation. Numer. Methods Partial Differ. Equations 35(2), 445\u2013477 (2019). https:\/\/doi.org\/10.1002\/num.22306","journal-title":"Numer. Methods Partial Differ. Equations"},{"issue":"5","key":"1987_CR56","doi-asserted-by":"publisher","first-page":"716","DOI":"10.1080\/10236198.2019.1624737","volume":"25","author":"RK Mohanty","year":"2019","unstructured":"Mohanty, R.K., Manchanda, G., Khan, A.: Compact half step approximation in exponential form for 2D second order quasi-linear elliptic partial differential equations. J. Differ. Equations Appl. 25(5), 716\u2013749 (2019). https:\/\/doi.org\/10.1080\/10236198.2019.1624737","journal-title":"J. Differ. Equations Appl."},{"key":"1987_CR57","doi-asserted-by":"publisher","first-page":"973","DOI":"10.1007\/s10910-024-01577-w","volume":"62","author":"RK Mohanty","year":"2024","unstructured":"Mohanty, R.K., Sharma, D.: A new 2-level implicit high accuracy compact exponential approximation for the numerical solution of nonlinear fourth order Kuramoto-Sivashinsky and Fisher-Kolmogorov equations. J. Math. Chem. 62, 973\u20131011 (2024). https:\/\/doi.org\/10.1007\/s10910-024-01577-w","journal-title":"J. Math. Chem."},{"key":"1987_CR58","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.1007\/s10910-023-01451-1","volume":"61","author":"RK Mohanty","year":"2023","unstructured":"Mohanty, R.K., Sharma, D.: A new 2-level compact off-step implicit method in exponential form for the solution of fourth order nonlinear parabolic equations. J. Math. Chem. 61, 1165\u20131204 (2023). https:\/\/doi.org\/10.1007\/s10910-023-01451-1","journal-title":"J. Math. Chem."},{"key":"1987_CR59","unstructured":"Jain, M.K.: Numerical Solution of Differential Equations: Finite Difference and Finite Element Methods. New AGE International Publishers, 4th Edit., New Delhi (2018)"},{"issue":"1","key":"1987_CR60","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1108\/HFF-10-2019-0731","volume":"31","author":"M Dehghan","year":"2021","unstructured":"Dehghan, M., Mohammadi, V.: The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations. Int. J. Numer. Methods Heat Fluid Flow 31(1), 106\u2013133 (2021). https:\/\/doi.org\/10.1108\/HFF-10-2019-0731","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"1987_CR61","doi-asserted-by":"publisher","first-page":"316","DOI":"10.1016\/j.enganabound.2022.06.024","volume":"143","author":"V Mohammadi","year":"2022","unstructured":"Mohammadi, V., Dehghan, M.: A POD-RBF-FD scheme for simulating chemotaxis models on surfaces. Eng. Anal. Boundary Elem. 143, 316\u2013330 (2022). https:\/\/doi.org\/10.1016\/j.enganabound.2022.06.024","journal-title":"Eng. Anal. Boundary Elem."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01987-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-024-01987-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-024-01987-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:27:04Z","timestamp":1760164024000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-024-01987-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,4]]},"references-count":61,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["1987"],"URL":"https:\/\/doi.org\/10.1007\/s11075-024-01987-6","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2024,12,4]]},"assertion":[{"value":"2 July 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 December 2024","order":3,"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 no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"This article does not contain any study with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}]}}