{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T22:48:49Z","timestamp":1780613329696,"version":"3.54.1"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,20]],"date-time":"2021-08-20T00:00:00Z","timestamp":1629417600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research, Developement and Innovation Fund of Hungary","award":["Project No. 129257, K_18"],"award-info":[{"award-number":["Project No. 129257, K_18"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, we construct novel numerical algorithms to solve the heat or diffusion equation. We start with 105 different leapfrog-hopscotch algorithm combinations and narrow this selection down to five during subsequent tests. We demonstrate the performance of these top five methods in the case of large systems with random parameters and discontinuous initial conditions, by comparing them with other methods. We verify the methods by reproducing an analytical solution using a non-equidistant mesh. Then, we construct a new nontrivial analytical solution containing the Kummer functions for the heat equation with time-dependent coefficients, and also reproduce this solution. The new methods are then applied to the nonlinear Fisher equation. Finally, we analytically prove that the order of accuracy of the methods is two, and present evidence that they are unconditionally stable.<\/jats:p>","DOI":"10.3390\/computation9080092","type":"journal-article","created":{"date-parts":[[2021,8,20]],"date-time":"2021-08-20T08:44:45Z","timestamp":1629449085000},"page":"92","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Stable, Explicit, Leapfrog-Hopscotch Algorithms for the Diffusion Equation"],"prefix":"10.3390","volume":"9","author":[{"given":"\u00c1d\u00e1m","family":"Nagy","sequence":"first","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1108-0099","authenticated-orcid":false,"given":"Issa","family":"Omle","sequence":"additional","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3901-3410","authenticated-orcid":false,"given":"Humam","family":"Kareem","sequence":"additional","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"},{"name":"Department of Mechanical Engineering, University of Technology, Baghdad 10066, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0439-3070","authenticated-orcid":false,"given":"Endre","family":"Kov\u00e1cs","sequence":"additional","affiliation":[{"name":"Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6206-3910","authenticated-orcid":false,"given":"Imre Ferenc","family":"Barna","sequence":"additional","affiliation":[{"name":"Wigner Research Center for Physics, 1051 Budapest, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4070-1376","authenticated-orcid":false,"given":"Gabriella","family":"Bognar","sequence":"additional","affiliation":[{"name":"Institute of Machine and Product Design, University of Miskolc, 3515 Miskolc, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"323","DOI":"10.35925\/j.multi.2020.4.36","article-title":"Construction and investigation of new numerical algorithms for the heat equation: Part 1","volume":"10","author":"Saleh","year":"2020","journal-title":"Multidiszcip. 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Appl., 26.","DOI":"10.3390\/mca26030061"},{"key":"ref_5","first-page":"1","article-title":"Benchmark solutions for three-dimensional transient heat transfer in two-dimensional environments via the time fourier transform","volume":"2","author":"Tadeu","year":"2005","journal-title":"Comput. Mater. Contin."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1016\/S0307-904X(99)00005-0","article-title":"Analytical solution of a spatially variable coefficient advection\u2013diffusion equation in up to three dimensions","volume":"23","author":"Zoppou","year":"1999","journal-title":"Appl. Math. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2469","DOI":"10.1002\/num.22730","article-title":"A class of new stable, explicit methods to solve the non-stationary heat equation","volume":"37","year":"2021","journal-title":"Numer. Methods Partial. Differ. Equ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1137\/0113044","article-title":"Nonsymmetric Difference Equations","volume":"13","author":"Gordon","year":"1965","journal-title":"J. Soc. Ind. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1093\/imamat\/6.4.375","article-title":"Hopscotch: A Fast Second-order Partial Differential Equation Solver","volume":"6","author":"Gourlay","year":"1970","journal-title":"IMA J. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1093\/imamat\/7.2.216","article-title":"General Hopscotch Algorithm for the Numerical Solution of Partial Differential Equations","volume":"7","author":"Gourlay","year":"1971","journal-title":"IMA J. Appl. Math."},{"key":"ref_11","unstructured":"(2021, July 11). Leapfrog Integration. Available online: https:\/\/en.wikipedia.org\/wiki\/Leapfrog_integration."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hockney, R.W., and Eastwood, J.W. (1989). Computer Simulation Using Particles, Taylor & Francis.","DOI":"10.1201\/9781439822050"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1063\/1.881812","article-title":"Understanding Molecular Simulation: From Algorithms to Applications","volume":"50","author":"Frenkel","year":"1997","journal-title":"Phys. Today"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1093\/imanum\/6.4.381","article-title":"Generalized Leapfrog Methods","volume":"6","author":"Iserles","year":"1986","journal-title":"Ima. J. Numer. Anal."},{"key":"ref_15","unstructured":"Hirsch, C. (1988). Numerical Computation of Internal and External Flows, Volume 1: Fundamentals of Numerical Discretization, Wiley."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1137\/S0036142994276979","article-title":"Stability Analysis of an Odd\u2014Even-Line Hopscotch Method for Three-Dimensional Advection\u2014Diffusion Problems","volume":"34","author":"Verwer","year":"1997","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Holmes, M.H. (2007). Introduction to Numerical Methods in Differential Equations, Springer.","DOI":"10.1007\/978-0-387-68121-4"},{"key":"ref_18","unstructured":"Munka, M., and P\u00e1pay, J. (2001). 4D Numerical Modeling of Petroleum Reservoir Recovery, Akad\u00e9miai Kiad\u00f3."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3","DOI":"10.32973\/jcam.2020.001","article-title":"New stable, explicit, first order method to solve the heat conduction equation","volume":"15","year":"2020","journal-title":"J. Comput. Appl. Mech."},{"key":"ref_20","first-page":"176","article-title":"Explicit-in-time goal-oriented adaptivity","volume":"347","author":"Calo","year":"2018","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_21","unstructured":"(2021, July 30). Heun\u2019s Method\u2014Wikipedia. Available online: https:\/\/en.wikipedia.org\/wiki\/Heun%27s_method."},{"key":"ref_22","unstructured":"M\u00e1ty\u00e1s, L., and Barna, I.F. (2021). General self-similar solutions of diffusion equation and related constructions. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Sedov, L.I. (2018). Similarity and Dimensional Methods in Mechanics, CRC Press.","DOI":"10.1201\/9780203739730"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"375210","DOI":"10.1088\/1751-8113\/43\/37\/375210","article-title":"Heat conduction: A telegraph-type model with self-similar behavior of solutions","volume":"43","author":"Barna","year":"2010","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3846\/mma.2020.10459","article-title":"Analytic self-similar solutions of the kardar-parisi-zhang interface growing equation with various noise terms","volume":"25","author":"Barna","year":"2020","journal-title":"Math. Model. Anal."},{"key":"ref_26","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2011). NIST Handbook of Mathematical Functions, Cambridge University Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","article-title":"The wave of advance of advantageous genes","volume":"7","author":"Fisher","year":"1937","journal-title":"Ann. Eugen."},{"key":"ref_28","first-page":"1","article-title":"A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem","volume":"1","author":"Kolmogorov","year":"1937","journal-title":"Bull. Mosc. Univ. Math. Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1007\/s00285-020-01547-1","article-title":"Travelling wave solutions in a negative nonlinear diffusion\u2013reaction model","volume":"81","author":"Li","year":"2020","journal-title":"J. Math. Biol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s12043-011-0243-8","article-title":"A highly accurate method to solve Fisher\u2019s equation","volume":"78","author":"Bastani","year":"2012","journal-title":"Pramana. J. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1186\/s13662-019-2080-x","article-title":"On the numerical solution of Fisher\u2019s equation with coefficient of diffusion term much smaller than coefficient of reaction term","volume":"2019","author":"Agbavon","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hiriart-Urruty, J.-B., and Lemar\u00e9chal, C. (2001). Fundamentals of Convex Analysis, Springer.","DOI":"10.1007\/978-3-642-56468-0"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/9\/8\/92\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:48:07Z","timestamp":1760165287000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/9\/8\/92"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,20]]},"references-count":32,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["computation9080092"],"URL":"https:\/\/doi.org\/10.3390\/computation9080092","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,20]]}}}