Implicit Exponentially Fitted Runge-Kutta Methods for Initial Value Problems

Abstract

This study introduces an enhanced implicit exponentially fitted Runge-Kutta (EFRK) method for the efficient solution of first-order initial value problems (IVPs) with exponential characteristics. The proposed approach minimizes the propagation of internal-stage errors to the final-stage error by deriving new coefficients without relying on frequency-evaluation algorithms. Unlike previous works that mainly addressed specific schemes, this research focuses on the general formulation of implicit EFRK methods. To ensure the conclusion is general and independent of the reference set, several cases of internal and external reference sets are considered. Numerical experiments are conducted to compare the proposed methods with existing ones through linear, nonlinear, and stiff test problems. Additionally, the stability analysis of both the proposed and the standard methods is conducted. The results reveal a larger stability region for the proposed methods, a more accurate solution when applied to stiff problems, and comparable errors for non-stiff problems compared to the standard methods.

Keywords

Implicit Runge-Kutta (RK) methods / Initial value problems (IVPs) / Stiff problems / Exponentially fitted (EF) methods / Stability enhancement / Error reduction / 65L04 / 65L06 / 65L20

References

[1] Abhulimen CE, Ukpebor LA. A new class of third derivative fourth-step exponential fitting numerical integrator for stiff differential equations. Int. J. Math. Sci. Optim.: Theor. Appl., 2018, 4(1): 382-391

[2] Burrage K, Cardone A, D'Ambrosio R, Paternoster B. Numerical solution of time fractional diffusion systems. Appl. Numer. Math., 2017, 116: 82-94

[3] Butcher JC. Numerical Methods for Ordinary Differential Equations, 2016John Wiley & Sons

[4] Cardone A. Stability analysis of EF Gaussian direct quadrature methods for Volterra integral equations. Appl. Numer. Math., 2023, 181: 241-251

[5] Cardone A, D'Ambrosio R, Paternoster B. Exponentially fitted IMEX methods for advection–diffusion problems. J. Comput. Appl. Math., 2017, 316: 100-108

[6] Cardone A, D'Ambrosio R, Paternoster B. High order exponentially fitted methods for Volterra integral equations with periodic solution. Appl. Numer. Math., 2017, 114: 18-29

[7] Cardone A, Ixaru LG, Paternoster B, Santomauro G. Ef-Gaussian direct quadrature methods for Volterra integral equations with periodic solution. Math. Comput. Simul, 2015, 110: 125-143

[8] Coleman JP, Ixaru LG. Truncation errors in exponential fitting for oscillatory problems. SIAM J. Numer. Anal., 2006, 44(4): 1441-1465

[9] Conte, D., D’Ambrosio, R., Giordano, G., Paternoster, B.: Regularized exponentially fitted methods for oscillatory problems. Journal of Physics: Conference Series 1564, 012013 (2020)

[10] Conte D, D’Ambrosio R, Moccaldi M, Paternoster B. Adapted explicit two-step peer methods. J. Numer. Math., 2017, 25(2): 125-145

[11] Conte D, Ixaru LG, Paternoster B, Santomauro G. Exponentially-fitted Gauss-Laguerre quadrature rule for integrals over an unbounded interval. J. Comput. Appl. Math., 2014, 255: 725-736

[12] Conte D, Paternoster B, Santomauro G. Exponential fitting quadrature rule for functional equations. AIP Conf. Proc., 2012, 1479(1): 1169-1172

[13] Conte D, Paternoster B, Santomauro G. An exponentially fitted quadrature rule over unbounded intervals. AIP Conf. Proc., 2012, 1479(1): 1173-1176

[14] D’Ambrosio R, Esposito E, Paternoster B. Exponentially fitted two-step hybrid methods for y′′=f(x,y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$y'' = f(x, y)$\end{document}. J. Comput. Appl. Math., 2011, 235(16): 4888-4897

[15] D’Ambrosio R, Esposito E, Paternoster B. Exponentially fitted two-step Runge-Kutta methods: construction and parameter selection. Appl. Math. Comput., 2012, 218(14): 7468-7480

[16] D’Ambrosio R, Ferro M, Paternoster B. Two-step hybrid collocation methods for y′′=f(x,y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$y'' = f (x, y)$\end{document}. Appl. Math. Lett., 2009, 22(7): 1076-1080

[17] D’Ambrosio R, Ferro M, Paternoster B. Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations. Math. Comput. Simul, 2011, 81(5): 1068-1084

[18] D'Ambrosio R, Ixaru LG, Paternoster B. Construction of the EF-based Runge-Kutta methods revisited. Comput. Phys. Commun., 2011, 182(2): 322-329

[19] D’Ambrosio R, Moccaldi M, Paternoster B. Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl., 2017, 74(5): 1029-1042

[20] D’Ambrosio R, Moccaldi M, Paternoster B. Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems. Comput. Phys. Commun., 2018, 226: 55-66

[21] D’Ambrosio R, Paternoster B. Exponentially fitted singly diagonally implicit Runge-Kutta methods. J. Comput. Appl. Math., 2014, 263: 277-287

[22] D’Ambrosio R, Paternoster B. Numerical solution of a diffusion problem by exponentially fitted finite difference methods. Springerplus, 2014, 3(1 425

[23] Fang Y, Lv H, You X. Novel exponentially fitted two-derivative Runge-Kutta methods for solving the radial Schrödinger equation. J. Math. Chem., 2025, 63(2): 546-577

[24] Franco JM, Rández L. Explicit exponentially fitted two-step hybrid methods of high order for second-order oscillatory IVPs. Appl. Math. Comput., 2016, 273: 493-505

[25] Ghawadri, N., Senu, N., Ismail, F., Ibrahim, Z.B.: Exponentially-fitted forth-order explicit modified Runge-Kutta type method for solving third-order ODEs. Journal of Physics: Conference Series 1132(1), 012016 (2018)

[26] Ghawadri N, Senu N, Ismail F, Ibrahim ZB. Exponentially fitted and trigonometrically fitted explicit modified Runge‐Kutta type methods for solving y′′′(x)=f(x,y,y′)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$y'''(x) = f (x, y, y')$\end{document}. J. Appl. Math., 2018, 2018(14029371

[27] Hollevoet, D.: Application of exponential fitting techniques to numerical methods for solving differential equations. Doctoral dissertation. Ghent University, Ghent (2013)

[28] Hollevoet D, Van Daele M, Berghe GV. The optimal exponentially-fitted Numerov method for solving two-point boundary value problems. J. Comput. Appl. Math., 2009, 230(1): 260-269

[29] Hollevoet D, Van Daele M, Berghe GV. Exponentially fitted methods applied to fourth-order boundary value problems. J. Comput. Appl. Math., 2011, 235(18): 5380-5393

[30] Ibrahimov VR. Exponentially fitted fifth-order two-step peer explicit methods. AIP Conf. Proc., 2017, 1863560029

[31] Ixaru LG. Operations on oscillatory functions. Comput. Phys. Commun., 1997, 105(1): 1-19

[32] Ixaru LG. Exponential fitting for interpolation of oscillatory functions. A numerical approach. J. Comput. Appl. Math., 2024, 437 115479

[33] Ixaru, L.G., Vanden Berghe, G.: Mathematical properties. In: Exponential Fitting. Mathematics and Its Applications, vol. 568. Springer, Dordrecht (2004)

[34] Kim KJ, Choi SH. Frequency-dependent interpolation rules using first derivatives for oscillatory functions. J. Comput. Appl. Math., 2007, 205(1): 149-160

[35] Marušić M. High-order exponentially fitted difference schemes for singularly perturbed two-point boundary value problems. Electron. Trans. Numer. Anal., 2018, 48: 329-347

[36] Ranjan R, Prasad HS. A novel exponentially fitted finite difference method for a class of 2nd order singularly perturbed boundary value problems with a simple turning point exhibiting twin boundary layers. J. Ambient. Intell. Humaniz. Comput., 2022, 13(9): 4207-4221

[37] Shokri A, Vigo-Aguiar J, Khalsaraei MM, Garcia-Rubio R. A new class of two-step P-stable TFPL methods for the numerical solution of second-order IVPs with oscillating solutions. J. Comput. Appl. Math., 2019, 354: 551-561

[38] Simos TE. A four-step exponentially fitted method for the numerical solution of the Schrödinger equation. J. Math. Chem., 2006, 40: 305-318

[39] Tang C, Yan H, Zhang H, Chen Z, Liu M, Zhang G. The arbitrary order implicit multistep schemes of exponential fitting and their applications. J. Comput. Appl. Math., 2005, 173(1): 155-168

[40] Tiwari S, Pandey RK. Exponentially-fitted pseudo Runge-Kutta method. Int. J. Comput. Sci. Math., 2020, 12(2): 105-116

[41] Tiwari S, Pandey RK. Revised version of exponentially fitted pseudo-Runge-Kutta method. Int. J. Comput. Sci. Math., 2021, 13(2): 116-125

[42] Van Daele M, Berghe GV, Vyver HV. Exponentially fitted quadrature rules of Gauss type for oscillatory integrands. Appl. Numer. Math., 2005, 53(2-4): 509-526

[43] Van de Vyver H. Frequency evaluation for exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math., 2005, 184(2): 442-463

[44] Van de Vyver, H. Phase properties of exponentially-fitted symmetric multistep methods for y′′=f(x,y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$y'' = f(x, y)$\end{document}. arXiv:math/0611801 (2006)

[45] Vanden Berghe G, De Meyer H, Van Daele M, Van Hecke T. Exponentially fitted Runge-Kutta methods. J. Comput. Appl. Math., 2000, 125(1-2): 107-115

[46] Berghe GV, Ixaru LG, De Meyer H. Frequency determination and step-length control for exponentially-fitted Runge–Kutta methods. J. Comput. Appl. Math., 2001, 132(1): 95-105

[47] Vanden BG, Ixaru LG, Van Daele M. Optimal implicit exponentially-fitted Runge-Kutta methods. Comput. Phys. Commun., 2001, 140(3): 346-357

[48] Wen J, Xiao A, Huang C. Multistep Runge-Kutta methods for Volterra integro-differential equations. J. Comput. Appl. Math., 2023, 426115384

[49] Wusu A. A two-parameter family of exponentially-fitted Obrechkoff methods for second-order boundary value problems. J. Res. Rev. Sci., 2020, 7: 28-32

[50] Zahra WK, Nasr MA. Exponentially fitted methods for solving two-dimensional time fractional damped Klein-Gordon equation with nonlinear source term. Commun. Nonlinear Sci. Numer. Simul., 2019, 73: 177-194

[51] Zahra WK, Nasr MA, Van Daele M. Exponentially fitted methods for solving time fractional nonlinear reaction-diffusion equation. Appl. Math. Comput., 2019, 358: 468-490

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