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A new mesh selection strategy with stiffness detection for explicit Runge-Kutta methods. (English) Zbl 1338.68293
Summary: In this paper, we develop a new mesh selection strategy based on the computation of some conditioning parameters which allows to give information about the conditioning and the stiffness of the problem. The reliability of the proposed algorithm is demonstrated by some numerical experiments. We observe that “when an initial value problem is run on a computer, the results may appear plausible even if they are unreliable because of some unrecognized numerical instability” [R. H. Miller, J. Comput. Phys. 2, 1–7 (1967; Zbl 0158.15501)]. The additional information about the behavior of the numerical solution provided by the new mesh selection algorithm are, therefore, of great interest for potential users of a numerical computer code.

MSC:
68W25 Approximation algorithms
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[1] Shampine, L. F., Stiffness and nonstiff differential equation solvers. II. detecting stiffness with Runge-Kutta methods, ACM Trans. Math. Software, 3, 1, 44-53, (1977) · Zbl 0349.65043
[2] Rentrop, P., Partitioned Runge-Kutta methods with stiffness detection and stepsize control, Numer. Math., 47, 4, 545-564, (1985) · Zbl 0625.65059
[3] Shampine, L. F., Diagnosing stiffness for Runge-Kutta methods, SIAM J. Sci. Statist. Comput., 12, 2, 260-272, (1991) · Zbl 0724.65073
[4] Ekeland, K.; Owren, B.; Øines, E., Stiffness detection and estimation of dominant spectra with explicit Runge-Kutta methods, ACM Trans. Math. Software, 24, 4, 368-382, (1998) · Zbl 0934.65068
[5] Sofroniou, M.; Spaletta, G., Construction of explicit Runge-Kutta pairs with stiffness detection, Math. Comput. Modelling, 40, 11-12, 1157-1169, (2004) · Zbl 1074.65087
[6] Hairer, E.; Wanner, G., (Solving ordinary differential equations. II. Stiff and differential-algebraic problems Springer Series in Computational Mathematics, vol. 14, (2010), Springer-Verlag Berlin), paperback · Zbl 1192.65097
[7] Sofroniou, M.; Spaletta, G., Extrapolation methods in Mathematica, JNAIAM J. Numer. Anal. Ind. Appl. Math., 3, 1-2, 105-121, (2008) · Zbl 1154.65062
[8] Petzold, L., Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations, SIAM J. Sci. Statist. Comput., 4, 1, 137-148, (1983) · Zbl 0518.65051
[9] Butcher, J. C., Order, stepsize and stiffness switching, Computing, 44, 3, 209-220, (1990) · Zbl 0719.65059
[10] Shampine, L. F.; Thompson, S., Stiff systems, Scholarpedia, 2, 3, 2855, (2007)
[11] Brugnano, L.; Trigiante, D., On the characterization of stiffness for odes, Dynam. Contin. Discrete Impuls. Systems, 2, 3, 317-335, (1996) · Zbl 0873.65071
[12] Brugnano, L.; Trigiante, D., Solving differential problems by multistep initial and boundary value methods, Stability and Control: Theory, Methods and Applications, vol. 6, (1998), Gordon and Breach Science Publishers Amsterdam · Zbl 0934.65074
[13] Iavernaro, F.; Mazzia, F.; Trigiante, D., Stability and conditioning in numerical analysis, JNAIAM J. Numer. Anal. Ind. Appl. Math., 1, 1, 91-112, (2006) · Zbl 1108.65041
[14] Brugnano, L.; Mazzia, F.; Trigiante, D., Fifty years of stiffness, (Recent Advances in Computational and Applied Mathematics, (2011), Springer Dordrecht), 1-21 · Zbl 1216.65083
[15] Cash, J. R.; Mazzia, F., Conditioning and hybrid mesh selection algorithms for two point boundary value problems, Scalable Computing: Practice and Experience, 10, 4, 347-361, (2009)
[16] Cash, J. R.; Mazzia, F.; Sumarti, N.; Trigiante, D., The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems, J. Comput. Appl. Math., 185, 2, 212-224, (2006) · Zbl 1077.65084
[17] Cash, J. R.; Hollevoet, D.; Mazzia, F.; Nagy, A. M., Algorithm 927: the MATLAB code bvptwp.m for the numerical solution of two point boundary value problems, ACM Trans. Math. Software, 39, 2, (2013) · Zbl 1295.65142
[18] Mazzia, F.; Trigiante, D., A hybrid mesh selection strategy based on conditioning for boundary value ODE problems, Numer. Algorithms, 36, 2, 169-187, (2004) · Zbl 1050.65072
[19] K. Soetaert, J.R. Cash, F. Mazzia, bvpSolve: Solvers for Boundary Value Problems of Ordinary Differential Equations, R package version 1.2.4 (2013). <http://CRAN.R-project.org/package=bvpSolve>.
[20] Mazzia, F.; Cash, J. R.; Soetaert, K., Solving boundary value problems in the open source software R: package bvpsolve, Opuscula Mathematica, 34, 2, 387-403, (2014) · Zbl 1293.65104
[21] Cash, J. R.; Mazzia, F., A new mesh selection algorithm, based on conditioning, for two-point boundary value codes, J. Comput. Appl. Math., 184, 2, 362-381, (2005) · Zbl 1076.65065
[22] Cash, J. R.; Mazzia, F., Hybrid mesh selection algorithms based on conditioning for two-point boundary value problems, JNAIAM J. Numer. Anal. Ind. Appl. Math., 1, 1, 81-90, (2006) · Zbl 1108.65082
[23] Miller, R. H., An experimental method for testing numerical stability in initial-value problems, J. Comput. Phys., 2, 1-7, (1967) · Zbl 0158.15501
[24] Mazzia, F.; Nagy, A. M., Stiffness detection strategy for explicit Runge Kutta methods, AIP Conf. Proc., 1281, 1, 239-242, (2010)
[25] Mazzia, F.; Magherini, C., (Test Set for Initial Value Problem Solvers, release 2.4, (2008), Department of Mathematics, University of Bari and INdAM, Research Unit of Bari)
[26] Ascher, U. M.; Mattheij, R. M.M.; Russell, R. D., Numerical solution of boundary value problems for ordinary differential equations, (Classics in Applied Mathematics, vol. 13, (1995), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA), corrected reprint of the 1988 original · Zbl 0671.65063
[27] Cash, J. R.; Karp, A. H., A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software, 16, 3, 201-222, (1990) · Zbl 0900.65234
[28] K. Soetaert, J.R. Cash, F. Mazzia, deTestSet: Testset for differential equations, R package version 1.1.1 (2013). <http://CRAN.R-project.org/package=deSolve>. · Zbl 1246.65104
[29] Mazzia, F.; Cash, J. R.; Soetaert, K., A test set for stiff initial value problem solvers in the open source software R: package detestset, J. Comput. Appl. Math., 236, 16, 4119-4131, (2012) · Zbl 1246.65104
[30] Soetaert, K.; Cash, J.; Mazzia, F., Solving differential equations in R, Use R!, (2012), Springer New York · Zbl 1252.65138
[31] C. Moler, MATLAB News & Notes, Stiff Differential Equations, 2003. <http://www.mathworks.com/company/newsletters/newsnotes/clevescorner/may03cleve.html>.
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