DOI: 10.12732/ijam.v37i6.6
REVISITING FINITE DIFFERENCE
SOLUTIONS FOR HEAT-TYPE EQUATIONS.
PART I: DIFFUSIVE TRANSFER
G. Garrido 1,§, J. L. G. Pestana 2
1 Departamento de Inform´atica
Universidad de Jaen, Campus Las Lagunillas
23071 Ja´en, SPAIN
2 Departamento de Fisica
Universidad de Jaen, Campus Las Lagunillas
Abstract. In this first paper of a series, we revisit all major systematic uncertainties that affect a complete and unbiased sample of five finite difference schemes for diffusion-like equations. In order to provide the coherent picture, unlike the existing way, we use as the key tenets both the reverse Taylor’s analysis and the discrete Fourier’s analysis, as well as the monotonicity analysis. For every type of scheme, their theoretical uncertainties are examined. A detailed graphical investigation is also provided and used to give a physical reinterpretation of the Courant-Friedrichs-Lewy-type condition. We find that no scheme considered in this study resolves the smaller length scales well. Furthermore, we present several numerical experiments on an equal footing corroborating our demonstrations and proving whether the accuracy of each scheme is impaired by the discontinuities in the data. A comparison with each other is made as well. Our results indicate that the simplest Schmidt’s scheme is also preferred by experiments.
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DOI: 10.12732/ijam.v37i6.6
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 6
References
[1] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM (2007).
[2] L. Peter Roed, Atmospheres and Oceans on Computers, Springer (2019).
[3] T.K. Sengupta, High Accuracy Computing Methods: Fluid Flows and Wave Phenomena, Cambridge University Press (2013).
[4] N.D. Katopodes, Free-Surface Flow Computational Methods, Elsevier (2019).
[5] D.J. Duffy, Finite Difference Methods in Financial Engineering, Wiley (2006).
[6] P. Sagaut, V.K. Suman, P. Sundaram, M.K. Rajpoot, Y.G. Bhumkar, S. Sengupta, A. Sengupta and T.K. Sengupta, Global spectral analysis: Review of numerical methods, Computers and Fluids, 261 (2023), 105915; doi: 10.1016/j.compfluid.2023.105915
[7] C.W. Hirt, Heuristic stability theory for finite-difference equations, Journal of Computational Physics, 2, No 4 (1968), 339–355; doi: 10.1016/0021-9991(68)90041-7
[8] J. von Neumann and R.D. Richtmyer, On the Numerical Solution of Partial Differential Equations of Parabolic Type, Los Alamos Technical Reports, LA–657 (1947).
[9] S.K. Godunov, Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics, Matematiceskij Sbornik, 47, No 89 (1959), 271–306.
[10] R.F. Warming and B.J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, Journal of Computational Physics, 14, No 2 (1974), 159–179; doi:10.1016/0021-9991(74)90011-4
[11] Y.-K. Kwok, Stability analysis of six-point
finite difference schemes for the constant coefficient convective-diffusion
equation, Computers Math. Applic., 23, No 12 (1992), 3–11;
doi: 10.1016/0898-1221(92)90088-Y
[12] I. Winnicki, J. Jasinski and S. Pietrek, New approach to the Lax-Wendroff modified differential equation for linear and nonlinear advection, Numer. Methods Partial Differential Eq., 35 (2019), 2275–2304; doi: 10.1002/num.22412
[13] M. Karam, J.C. Sutherland and T. Saad, PyModPDE: A python software for modified equation analysis, SoftwareX, 12 (2020), 100541; doi: 10.1016/j.softx.2020.100541
[14] T. Bodnar, P. Fraunie and K. Kozel, Modified equation for a class of explicit and implicit schemes solving one-dimensional advection problem, Acta Polytechnica, 61 (2021), 49–58; doi: 10.14311/AP.2021.61.0049
[15] L.F. Richardson, The approximate arithmetical
solution by finite differences of physical problems involving differential
equations, with an application to the stresses in a masonry dam, Philosophical
Transactions of the Royal Society of London, Series A, 210 (1910), 307–357;
doi: 10.1098/rsta.1911.0009
[16] E. Schmidt, Uber die Anwendung der
Differenzenrechnung auf technische Anheiz- und Abkuhlungsprobleme, Beitrage zur
Technischen Mechanik und Technischen Physik: August Foppl zum Siebzigsten
Geburtstag am 25. Januar 1924 (1924), 179–189;
doi: 10.1007/978-3-642-51983-3 19
[17] R. Courant, K. Friedrichs and H. Lewy, Uber die
partiellen Differenzengleichungen der mathematischen Physik, Mathematische
Annalen, 100 (1928), 32–74;
doi: 10.1147/rd.112.0215
[18] C.F. Curtiss and J.O. Hirschfelder, Integration of Stiff Equations, Proc. Natl. Acad. Sci., 38 (1952), 235–243; doi: 10.1073/pnas.38.3.235
[19] J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Mathematical Proceedings of the Cambridge Philosophical Society, 43 (1947), 50–67; doi: 10.1017/S0305004100023197
[20] L.H. Thomas, Elliptic Problems in Linear Difference Equations over a Network, Watson Sci. Comput. Lab. Rept., Columbia University (1949).
[21] P. Laasonen, Uber eine Methode zur Losung der Warmeleitungsgleichung, Acta Mathematica, 81 (1949), 309–317; doi: 10.1007/BF02395025
[22] E.C. Du Fort and S.P. Frankel, Stability conditions in the numerical treatment of parabolic differential equations, Mathematical Tables and Other Aids to Computation, 7, No 43 (1953), 135–152; doi: 10.2307/2002754
[23] P.D. Lax and R.D. Richtmyer, Survey of the
stability of linear finite difference equations, Communications on Pure and
Applied Mathematics, 9, No 2 (1956), 267–293;
doi: 10.1002/cpa.3160090206
[24] J.B.J. Fourier, Joseph Fourier, 1768–1830, MIT Press (1972).
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