{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T13:19:45Z","timestamp":1778678385785,"version":"3.51.4"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2013,9,4]],"date-time":"2013-09-04T00:00:00Z","timestamp":1378252800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2014,6]]},"DOI":"10.1007\/s10915-013-9775-z","type":"journal-article","created":{"date-parts":[[2013,9,3]],"date-time":"2013-09-03T07:18:21Z","timestamp":1378192701000},"page":"626-666","source":"Crossref","is-referenced-by-count":11,"title":["Method of Moving Frames to Solve (An)isotropic Diffusion Equations on Curved Surfaces"],"prefix":"10.1007","volume":"59","author":[{"given":"Sehun","family":"Chun","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2013,9,4]]},"reference":[{"issue":"1","key":"9775_CR1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0021-9991(02)00057-8","volume":"185","author":"D Adalsteinsson","year":"2003","unstructured":"Adalsteinsson, D., Sethian, J.A.: Transport and diffusion of material quantities on propagating interfaces via level set methods. J. Comput. Phys. 185(1), 271\u2013288 (2003)","journal-title":"J. Comput. Phys."},{"issue":"13","key":"9775_CR2","doi-asserted-by":"crossref","first-page":"1816","DOI":"10.1161\/01.CIR.0000058461.86339.7E","volume":"107","author":"R Arora","year":"2003","unstructured":"Arora, R., Verheule, S., Scott, L., Navarrete, A., Katari, V., Wilson, E., Vaz, D., Olgin, J.E.: Arrhythmogenic substrate of the pulmonary veins assessed by high-resolution optical mapping. Circulation 107(13), 1816\u20131821 (2003)","journal-title":"Circulation"},{"key":"9775_CR3","unstructured":"Aslanidi, O.V., Colman, M.A., Zhao, J., Smaill, B.H., Gilbert, S.H., Hancox, J.C., Boyett, M.R., Zhang, H.: Arrhythmogenic substrate for atrial fibrillation: insights from an integrative computational model of pulmonary veins. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2012, 203\u2013206 (2012)"},{"key":"9775_CR4","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1137\/1036141","volume":"36","author":"I Babu\u0161ka","year":"1994","unstructured":"Babu\u0161ka, I., Suri, M.: The $$p$$ p and $$h-p$$ h - p versions of the finite element method, basic principles and properties. SIAM Rev. 36, 578\u2013632 (1994)","journal-title":"SIAM Rev."},{"issue":"4","key":"9775_CR5","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s00285-009-0315-2","volume":"61","author":"M Bergdorf","year":"2010","unstructured":"Bergdorf, M., Sbalzarini, I.F., Koumoutsakos, P.: A lagrangian particle method for reaction-diffusion systems on deforming surfaces. J. Math. Biol. 61(4), 649\u2013663 (2010)","journal-title":"J. Math. Biol."},{"issue":"17","key":"9775_CR6","doi-asserted-by":"crossref","first-page":"6514","DOI":"10.1016\/j.jcp.2009.05.046","volume":"228","author":"P-E Bernard","year":"2009","unstructured":"Bernard, P.-E., Remacle, J.-F., Comblen, R., Legat, V., Hillewaert, K.: High-order discontinuous Galerkin schemes on general 2D manifolds applied to shallow water equations. J. Comput. Phys. 228(17), 6514\u20136535 (2009)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"9775_CR7","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1006\/jcph.2001.6937","volume":"174","author":"M Bertalmio","year":"2001","unstructured":"Bertalmio, M., Cheng, L., Osher, S., Sapiro, G.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759\u2013780 (2001)","journal-title":"J. Comput. Phys."},{"key":"9775_CR8","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1017\/S0962492906220014","volume":"15","author":"JC Butcher","year":"2006","unstructured":"Butcher, J.C.: General linear methods. Acta Numer. 15, 157\u2013256 (2006)","journal-title":"Acta Numer."},{"key":"9775_CR9","volume-title":"Geometry of Riemannian Spaces","author":"\u00c9 Cartan","year":"2001","unstructured":"Cartan, \u00c9.: Geometry of Riemannian Spaces. Math Sci Press, Brookline, MA (2001)"},{"key":"9775_CR10","volume-title":"Riemannian Geometry in an Orthogonal Frame","author":"\u00c9 Cartan","year":"2002","unstructured":"Cartan, \u00c9.: Riemannian Geometry in an Orthogonal Frame. World Scientific Pub. Co. Inc., Singapore (2002)"},{"key":"9775_CR11","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1016\/j.hrthm.2007.08.017","volume":"4","author":"EM Cherry","year":"2007","unstructured":"Cherry, E.M., Ehrlich, J.R., Nattel, S., Fenton, F.H.: Pulmonary vein reentry-properties and size matter: insights from a computational analysis. Heart Rhythm 4, 1553\u20131562 (2007)","journal-title":"Heart Rhythm"},{"issue":"2","key":"9775_CR12","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/j.jcp.2004.06.005","volume":"201","author":"M Christensen","year":"2004","unstructured":"Christensen, M.: How to simulate anisotropic diffusion processes on curved surfaces. J. Comput. Phys. 201(2), 421\u2013438 (2004)","journal-title":"J. Comput. Phys."},{"key":"9775_CR13","doi-asserted-by":"crossref","first-page":"5171","DOI":"10.1063\/1.1597476","volume":"119","author":"M Christensen","year":"2003","unstructured":"Christensen, M., Pedersen, J.B.: Diffusion in inhomogeneous and anisotropic media. J. Chem. Phys. 119, 5171\u20135175 (2003)","journal-title":"J. Chem. Phys."},{"issue":"2","key":"9775_CR14","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1007\/s10915-011-9570-7","volume":"53","author":"S Chun","year":"2012","unstructured":"Chun, S.: Method of moving frames to solve conservation laws on curved surfaces. J. Sci. Comput. 53(2), 268\u2013294 (2012)","journal-title":"J. Sci. Comput."},{"key":"9775_CR15","unstructured":"Chun, S.: Geometric analysis on the unidirectionality of the PVs for atrial reentry. Submitted (2013)"},{"key":"9775_CR16","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1137\/070706616","volume":"47","author":"B Cockburn","year":"2009","unstructured":"Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319\u20131365 (2009)","journal-title":"SIAM J. Numer. Anal."},{"key":"9775_CR17","doi-asserted-by":"crossref","unstructured":"Desbrun, M., Meyer, M., Schr\u0151der, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. SIGGRAPH99, pp. 317\u2013324 (1999)","DOI":"10.1145\/311535.311576"},{"issue":"7","key":"9775_CR18","doi-asserted-by":"crossref","first-page":"571","DOI":"10.2307\/2320587","volume":"86","author":"M Eisenberg","year":"1979","unstructured":"Eisenberg, M., Guy, R.: A proof of the Hairy Ball theorem. Am. Math. Mon. 86(7), 571\u2013574 (1979)","journal-title":"Am. Math. Mon."},{"issue":"5","key":"9775_CR19","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.1137\/070680503","volume":"46","author":"M Ethier","year":"2008","unstructured":"Ethier, M., Bourgault, Y.: Semi-implicit and time-discretization schemes for the bidomain model. SIAM J. Numer. Anal. 46(5), 2443\u20132468 (2008)","journal-title":"SIAM J. Numer. Anal."},{"key":"9775_CR20","volume-title":"Partial Differential Equations","author":"LC Evans","year":"2000","unstructured":"Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, RI (2000)"},{"key":"9775_CR21","unstructured":"FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1(6), 445\u2013466 (1961)"},{"key":"9775_CR22","unstructured":"Geuzaine, C., Remacle, J.-F.: Gmash: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods. Eng. 79(11), 1309\u20131331 (2009)"},{"issue":"3","key":"9775_CR23","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s10915-005-9012-5","volume":"29","author":"JB Greer","year":"2006","unstructured":"Greer, J.B.: An improvement of a recent Eulerian method for solving PDEs on general geometries. J. Sci. Comput. 29(3), 321\u2013352 (2006)","journal-title":"J. Sci. Comput."},{"key":"9775_CR24","doi-asserted-by":"crossref","unstructured":"Greer, J.B., Bertozzi, A.L., Sapiro, G.: Fourth order partial differential equations on general geometries. UCLA CAM, Report 05(17) (2005)","DOI":"10.21236\/ADA524786"},{"key":"9775_CR25","unstructured":"Guggenheimer, H.W.: Differential Geometry. Dover Publications, Inc., New York (1963)"},{"issue":"10","key":"9775_CR26","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1056\/NEJM199809033391003","volume":"339","author":"M Hal\u030bssaguerre","year":"1998","unstructured":"Hal\u030bssaguerre, M., Jal\u030bs, P., Shah, D.C., Takahashi, A., Hocini, M., Quiniou, G., Garrigue, S., Le Mouroux, A., Le M\u017dtayer, P., Cl\u017dmenty, J.: Spontaneous initiation of atrial fibrillation by ectopic beats originating in the pulmonary veins. N. Engl. J. Med. 339(10), 659\u2013666 (1998)","journal-title":"N. Engl. J. Med."},{"key":"9775_CR27","unstructured":"Hall, J.E.: Guyton and Hall Textbook of Medical Physiology, 12th revised edition. Saunders, Philadelphia, PA (2010)"},{"key":"9775_CR28","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s10915-011-9501-7","volume":"51","author":"RM Kirby","year":"2012","unstructured":"Kirby, R.M., Sherwin, S.J., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51, 183\u2013212 (2012)","journal-title":"J. Sci. Comput."},{"key":"9775_CR29","unstructured":"Lipschultz, M.: Schaums Outlines: Differential Geometry. McGraw-Hill (2001)"},{"issue":"6","key":"9775_CR30","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1063\/1.1724316","volume":"4","author":"FK Manasse","year":"1963","unstructured":"Manasse, F.K., Misner, C.W.: Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4(6), 735 (1963)","journal-title":"J. Math. Phys."},{"key":"9775_CR31","doi-asserted-by":"crossref","unstructured":"Meyer, M., Desbrun, M., Schr\u0151der, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. VisMath \u201902 Proceedings (2002)","DOI":"10.1007\/978-3-662-05105-4_2"},{"key":"9775_CR32","unstructured":"Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. W. H. Freeman and Co, Ltd, New York (1973)"},{"key":"9775_CR33","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1023\/A:1005131729982","volume":"25","author":"VG Morozov","year":"1999","unstructured":"Morozov, V.G., Davydov, N.V., Davydov, V.A.: Propagation of curved activation fronts in anisotropic excitable media. J. Biol. Phys. 25, 87\u2013100 (1999)","journal-title":"J. Biol. Phys."},{"issue":"1","key":"9775_CR34","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1161\/CIRCEP.107.754564","volume":"1","author":"S Nattel","year":"2008","unstructured":"Nattel, S., Burstein, B., Dobrev, D.: Atrial remodeling and atrial fibrillation: mechanisms and implications. Circ. Arrhythm. Electrophysiol. 1(1), 62\u201373 (2008)","journal-title":"Circ. Arrhythm. Electrophysiol."},{"issue":"2","key":"9775_CR35","doi-asserted-by":"crossref","first-page":"1271","DOI":"10.1016\/j.jcp.2007.05.025","volume":"226","author":"IL Novak","year":"2007","unstructured":"Novak, I.L., Gao, F., Choi, Y.S., Resasco, D., Schaff, J.C., Slepchenko, B.M.: Diffusion on a curved surface coupled to diffusion in the volume: application to cell biology. J. Comput. Phys. 226(2), 1271\u20131290 (2007)","journal-title":"J. Comput. Phys."},{"key":"9775_CR36","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1109\/34.56205","volume":"12","author":"P Perona","year":"1990","unstructured":"Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629\u2013639 (1990)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"1","key":"9775_CR37","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.jcp.2005.08.021","volume":"213","author":"JA Pudykiewicz","year":"2006","unstructured":"Pudykiewicz, J.A.: Numerical solution of the reaction-advection-diffusion equation on the sphere. J. Comput. Phys. 213(1), 358\u2013390 (2006)","journal-title":"J. Comput. Phys."},{"issue":"8","key":"9775_CR38","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1109\/10.310090","volume":"41","author":"JM Rogers","year":"1994","unstructured":"Rogers, J.M., McCulloch, A.D.: A collocation-Galerkin finite element model of cardiac action potential propagation. IEEE Trans. Biomed. Eng. 41(8), 743\u2013757 (1994)","journal-title":"IEEE Trans. Biomed. Eng."},{"key":"9775_CR39","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511623783","volume-title":"The Laplacian on a Riemannian Manifold","author":"S Rosenberg","year":"1997","unstructured":"Rosenberg, S.: The Laplacian on a Riemannian Manifold. Cambridge University Press, Cambridge (1997)"},{"key":"9775_CR40","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511626319","volume-title":"Geometric Partial Differential Equations and Image Analysis","author":"G Sapiro","year":"2001","unstructured":"Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2001)"},{"issue":"3","key":"9775_CR41","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1529\/biophysj.105.073809","volume":"90","author":"IF Sbalzarini","year":"2006","unstructured":"Sbalzarini, I.F., Hayer, A., Helenius, A., Koumoutsakos, P.: Simulations of (an)isotropic diffusion on curved biological surfaces. Biophys. J. 90(3), 878\u2013885 (2006)","journal-title":"Biophys. J."},{"issue":"32","key":"9775_CR42","doi-asserted-by":"crossref","first-page":"11151","DOI":"10.1073\/pnas.0504953102","volume":"102","author":"P Schwartz","year":"2005","unstructured":"Schwartz, P., Adalsteinsson, D., Colella, P., Arkin, A.P., Onsum, M.: Numerical computation of diffusion on a surface. Proc. Natl. Acad. Sci. USA 102(32), 11151\u201311156 (2005)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"9775_CR43","unstructured":"Sherwin, S.J., Kirby, R.M., et al.: Nektar++: open source software library for spectral\/hp element method. Website: http:\/\/www.nektar.info"},{"key":"9775_CR44","volume-title":"A Comprehensive Introduction to Differential Geometry","author":"M Spivak","year":"2005","unstructured":"Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I, 3rd edn. Publish or Perish, INC, Berkeley, CA (2005)","edition":"3"},{"key":"9775_CR45","volume-title":"A Comprehensive Introduction to Differential Geometry","author":"M Spivak","year":"2005","unstructured":"Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 3rd edn. Publish or Perish, INC, Berkeley, CA (2005)","edition":"3"},{"key":"9775_CR46","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1023\/A:1008152115986","volume":"36","author":"B Tang","year":"2000","unstructured":"Tang, B., Sapiro, G., Caselles, V.: Diffusion of general data on non-flat manifolds via harmonic maps theory: the direction diffusion case. Int. J. Comput. Vis. 36, 149\u2013161 (2000)","journal-title":"Int. J. Comput. Vis."},{"key":"9775_CR47","doi-asserted-by":"crossref","unstructured":"Turing, A.M.: The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences. 237, 37\u201372 (1953)","DOI":"10.1098\/rstb.1952.0012"},{"key":"9775_CR48","unstructured":"Vos, E.J., Eskilsson, C., Bolis, A., Chun, S., Kirby, R.M., Sherwin, S.J.: A generic framework for time-stepping pdes: general linear methods, object-oriented implementation and application to fluid problems. Int. J. Comput. Fluid. Dyn. 25(3), 107\u2013125 (2011)"},{"key":"9775_CR49","volume-title":"An Introduction to Riemannian Geometry and the Tensor Calculus","author":"CE Weatherburn","year":"1957","unstructured":"Weatherburn, C.E.: An Introduction to Riemannian Geometry and the Tensor Calculus. Cambridge University Press, Cambridge (1957)"},{"issue":"8","key":"9775_CR50","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1161\/01.CIR.103.8.1157","volume":"103","author":"TJ Wu","year":"2001","unstructured":"Wu, T.J., Ong, J.J., Chang, C.M., Doshi, R.N., Yashima, M., Huang, H.L., FIshbein, M.C., Ting, C.T., Karagueuzian, H.S., Chen, P.S.: Pulmonary veins and ligament of marshall as sources of rapid activations in a canine model of sustained atrial fibrillation. Circulation 103(8), 1157\u20131163 (2001)","journal-title":"Circulation"},{"key":"9775_CR51","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1016\/j.camwa.2004.05.001","volume":"48","author":"G Xu","year":"2004","unstructured":"Xu, G.: Convergence of discrete Laplace\u2013Beltrami operators over surfaces. Comput. Math. Appl. 48, 347\u2013360 (2004)","journal-title":"Comput. Math. Appl."},{"issue":"34","key":"9775_CR52","doi-asserted-by":"crossref","first-page":"15063","DOI":"10.1073\/pnas.1008837107","volume":"107","author":"RJ Young","year":"2010","unstructured":"Young, R.J., Panfilov, A.V.: Anisotropy of wave propagation in the heart can be modeled by a Riemannian electrophysiological metric. Proc. Natl. Acad. Sci. USA 107(34), 15063\u201315068 (2010)","journal-title":"Proc. Natl. Acad. Sci. USA"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9775-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-013-9775-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-013-9775-z","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,23]],"date-time":"2019-07-23T00:55:52Z","timestamp":1563843352000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-013-9775-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,9,4]]},"references-count":52,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2014,6]]}},"alternative-id":["9775"],"URL":"https:\/\/doi.org\/10.1007\/s10915-013-9775-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,9,4]]}}}