Explicit general linear methods with a large stability region for Volterra integro-differential equations
Abstract
In this paper, we describe the construction of a class of methods with a large area of the stability region for solving Volterra integro-differential equations. In the structure of these methods which is based on a subclass of explicit general linear methods with and without Runge-Kutta stability property, we use an adequate quadrature rule to approximate the integral term of the equation. The free parameters of the methods are used to obtain methods with a large stability region. The efficiency of the proposed methods is verified with some numerical experiments and comparisons with other existing methods.
Keyword : Volterra integro-differential equations, general linear methods, Runge–Kutta stability, region of absolute stability, Gregory quadrature rule
How to Cite
Mahdi, H., Hojjati, G., & Abdi, A. (2019). Explicit general linear methods with a large stability region for Volterra integro-differential equations. Mathematical Modelling and Analysis, 24(4), 478-493. https://doi.org/10.3846/mma.2019.029
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Bra´s and A. Cardone. Construction of efficient general linear methods for non-stiff differential systems. Mathematical Modelling and Analysis, 17(2):171– 189, 2012. https://doi.org/10.3846/13926292.2012.655789
M. Bra´s, A. Cardone and R. D’Ambrosio. Implementation of explicit Nordsieck methods with inherent quadratic stability. Mathematical Modelling and Analysis, 18(2):289–307, 2013. https://doi.org/10.3846/13926292.2013.785039
H. Brunner. Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations. Mathematics of computation, 42(165):95–109, 1984. https://doi.org/10.2307/2007561
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J.C. Butcher. On the convergence of numerical solutions to ordinary differential equations. Math. Comp., 20(93):1–10, 1966. https://doi.org/10.1090/S0025-5718-1966-0189251-X
J.C. Butcher. Diagonally-implicit multi-stage integration methods. Applied Numerical Mathematics, 11(5):347–363, 1993. https://doi.org/10.1016/0168-9274(93)90059-Z
J.C. Butcher. Numerical methods for ordinary differential equations. Wiley, 2016.
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J.C. Butcher and Z. Jackiewicz. Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Applied Numerical Mathematics, 27(1):1–12, 1998. https://doi.org/10.1016/S0168-9274(97)00109-8
J.C. Butcher and Z. Jackiewicz. Construction of general linear methods with Runge–Kutta stability properties. Numer. Algor., 36(1):53–72, 2004. https://doi.org/10.1023/B:NUMA.0000027738.54515.50
J.C. Butcher and W.M. Wright. The construction of practical general linear methods. BIT, 43(4):695–721, 2003. https://doi.org/10.1023/B:BITN.0000009952.71388.23
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A. Cardone, D. Conte, R. D’Ambrosio and B. Paternoster. Collocation methods for Volterra integral and integro-differential equations: A review. Axioms, 7(3):45, 2018. https://doi.org/10.3390/axioms7030045
A. Cardone, D. Conte and B. Paternoster. A family of multistep collocation methods for Volterra integro-differential equations. In AIP Conference Proceedings, volume 1168, pp. 358–361. AIP, 2009. https://doi.org/10.1063/1.3241469
A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Numer. Algor., 60(1):1–25, 2012. https://doi.org/10.1007/s11075-011-9509-y
A. Cardone, Z. Jackiewicz and H. Mittelmann. Optimization-based search for Nordsieck methods of high order with quadratic stability polynomials. Mathematical Modelling and Analysis, 17(3):293–308, 2012. https://doi.org/10.3846/13926292.2012.685497
J. Chollom and Z. Jackiewicz. Construction of two-step Runge– Kutta methods with large regions of absolute stability. Journal of computational and applied mathematics, 157(1):125–137, 2003. https://doi.org/10.1016/S0377-0427(03)00382-0
D. Costarelli and R. Spigler. A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions. Journal of Integral Equations and Applications, 26(1):15–52, 2014. https://doi.org/10.1216/JIE-2014-26-1-15
S. Fazeli and G. Hojjati. Numerical solution of Volterra integro-differential equations by superimplicit multistep collocation methods. Numerical Algorithms, 68(4):741–768, 2015. https://doi.org/10.1007/s11075-014-9870-8
Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, 2009.
P. Linz. Analytical and numerical methods for Volterra equations, volume 7. SIAM, 1985.
C. Lubich. Runge-Kutta theory for Volterra integrodifferential equations. Numerische Mathematik, 40(1):119–135, 1982. https://doi.org/10.1007/BF01459081
H. Mahdi, A. Abdi and G. Hojjati. Efficient general linear methods for a class of Volterra integro-differential equations. Applied Numerical Mathematics, 127:95– 109, 2018. https://doi.org/10.1016/j.apnum.2018.01.001
A. Makroglou. Hybrid methods in the numerical solution of Volterra integrodifferential equations. IMA Journal of Numerical Analysis, 2(1):21–35, 1982. https://doi.org/10.1093/imanum/2.1.21
J. Matthys. A-stable linear multistep methods for Volterra integrodifferential equations. Numerische Mathematik, 27(1):85–94, 1976. https://doi.org/10.1007/BF01399087
B.P. Sommeijer, W. Couzy and P.J. van der Houwen. A-stable parallel block methods for ordinary and integro-differential equations. Applied numerical mathematics, 9(3-5):267–281, 1992. https://doi.org/10.1016/0168-9274(92)90021-5
Y. Wei and Y. Chen. Legendre spectral collocation method for neutral and highorder Volterra integro-differential equation. Applied Numerical Mathematics, 81:15–29, 2014. https://doi.org/10.1016/j.apnum.2014.02.012
M.A. Wolfe and G.M. Phillips. Some methods for the solution of non-singular Volterra integro-differential equations. The Computer Journal, 11(3):334–336, 1968. https://doi.org/10.1093/comjnl/11.3.334
P.H.M. Wolkenfelt. The construction of reducible quadrature rules for Volterra integral and integro-differential equations. IMA Journal of Numerical Analysis, 2(2):131–152, 1982. https://doi.org/10.1093/imanum/2.2.131
N. Barghi Oskouie, G. Hojjati and A. Abdi. Efficient second derivative methods with extended stability regions for non-stiff IVPs. Computational and Applied Mathematics, 37(2):2001–2016, 2018. https://doi.org/10.1007/s40314-017-0436-y
Z. Bartoszewski and Z. Jackiewicz. Explicit Nordsieck methods with extended stability regions. Applied Mathematics and Computation, 218(10):6056–6066, 2012. https://doi.org/10.1016/j.amc.2011.11.088
M.I. Berenguer, M.A. Fortes, A.I. Guillem-Garralda and M.R. Gal´an. Linear Volterra integro-differential equation and Schauder bases. Applied Mathematics and Computation, 159(2):495–507, 2004. https://doi.org/10.1016/j.amc.2003.08.132
M. Bra´s and A. Cardone. Construction of efficient general linear methods for non-stiff differential systems. Mathematical Modelling and Analysis, 17(2):171– 189, 2012. https://doi.org/10.3846/13926292.2012.655789
M. Bra´s, A. Cardone and R. D’Ambrosio. Implementation of explicit Nordsieck methods with inherent quadratic stability. Mathematical Modelling and Analysis, 18(2):289–307, 2013. https://doi.org/10.3846/13926292.2013.785039
H. Brunner. Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations. Mathematics of computation, 42(165):95–109, 1984. https://doi.org/10.2307/2007561
H. Brunner. Collocation methods for Volterra integral and related functional differential equations, volume 15. Cambridge University Press, 2004.
H. Brunner and J.D. Lambert. Stability of numerical methods for Volterra integro-differential equations. Computing, 12(1):75–89, 1974. https://doi.org/10.1007/BF02239501
H. Brunner and P.J. van der Houwen. The numerical solution of Volterra equations, volume 3. Elsevier Science Ltd, 1986.
J.C. Butcher. On the convergence of numerical solutions to ordinary differential equations. Math. Comp., 20(93):1–10, 1966. https://doi.org/10.1090/S0025-5718-1966-0189251-X
J.C. Butcher. Diagonally-implicit multi-stage integration methods. Applied Numerical Mathematics, 11(5):347–363, 1993. https://doi.org/10.1016/0168-9274(93)90059-Z
J.C. Butcher. Numerical methods for ordinary differential equations. Wiley, 2016.
J.C. Butcher and Z. Jackiewicz. Diagonally implicit general linear methods for ordinary differential equations. BIT Numerical Mathematics, 33(3):452–472, 1993. https://doi.org/10.1007/BF01990528
J.C. Butcher and Z. Jackiewicz. Construction of diagonally implicit general linear methods of type 1 and 2 for ordinary differential equations. Applied Numerical Mathematics, 21(4):385–415, 1996. https://doi.org/10.1016/S0168-9274(96)00043-8
J.C. Butcher and Z. Jackiewicz. Implementation of diagonally implicit multistage integration methods for ordinary differential equations. SIAM Journal on Numerical Analysis, 34(6):2119–2141, 1997. https://doi.org/10.1137/S0036142995282509
J.C. Butcher and Z. Jackiewicz. Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Applied Numerical Mathematics, 27(1):1–12, 1998. https://doi.org/10.1016/S0168-9274(97)00109-8
J.C. Butcher and Z. Jackiewicz. Construction of general linear methods with Runge–Kutta stability properties. Numer. Algor., 36(1):53–72, 2004. https://doi.org/10.1023/B:NUMA.0000027738.54515.50
J.C. Butcher and W.M. Wright. The construction of practical general linear methods. BIT, 43(4):695–721, 2003. https://doi.org/10.1023/B:BITN.0000009952.71388.23
A. Cardone and D. Conte. Multistep collocation methods for Volterra integrodifferential equations. Applied Mathematics and Computation, 221:770–785, 2013. https://doi.org/10.1016/j.amc.2013.07.012
A. Cardone, D. Conte, R. D’Ambrosio and B. Paternoster. Collocation methods for Volterra integral and integro-differential equations: A review. Axioms, 7(3):45, 2018. https://doi.org/10.3390/axioms7030045
A. Cardone, D. Conte and B. Paternoster. A family of multistep collocation methods for Volterra integro-differential equations. In AIP Conference Proceedings, volume 1168, pp. 358–361. AIP, 2009. https://doi.org/10.1063/1.3241469
A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Numer. Algor., 60(1):1–25, 2012. https://doi.org/10.1007/s11075-011-9509-y
A. Cardone, Z. Jackiewicz and H. Mittelmann. Optimization-based search for Nordsieck methods of high order with quadratic stability polynomials. Mathematical Modelling and Analysis, 17(3):293–308, 2012. https://doi.org/10.3846/13926292.2012.685497
J. Chollom and Z. Jackiewicz. Construction of two-step Runge– Kutta methods with large regions of absolute stability. Journal of computational and applied mathematics, 157(1):125–137, 2003. https://doi.org/10.1016/S0377-0427(03)00382-0
D. Costarelli and R. Spigler. A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions. Journal of Integral Equations and Applications, 26(1):15–52, 2014. https://doi.org/10.1216/JIE-2014-26-1-15
S. Fazeli and G. Hojjati. Numerical solution of Volterra integro-differential equations by superimplicit multistep collocation methods. Numerical Algorithms, 68(4):741–768, 2015. https://doi.org/10.1007/s11075-014-9870-8
Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, 2009.
P. Linz. Analytical and numerical methods for Volterra equations, volume 7. SIAM, 1985.
C. Lubich. Runge-Kutta theory for Volterra integrodifferential equations. Numerische Mathematik, 40(1):119–135, 1982. https://doi.org/10.1007/BF01459081
H. Mahdi, A. Abdi and G. Hojjati. Efficient general linear methods for a class of Volterra integro-differential equations. Applied Numerical Mathematics, 127:95– 109, 2018. https://doi.org/10.1016/j.apnum.2018.01.001
A. Makroglou. Hybrid methods in the numerical solution of Volterra integrodifferential equations. IMA Journal of Numerical Analysis, 2(1):21–35, 1982. https://doi.org/10.1093/imanum/2.1.21
J. Matthys. A-stable linear multistep methods for Volterra integrodifferential equations. Numerische Mathematik, 27(1):85–94, 1976. https://doi.org/10.1007/BF01399087
B.P. Sommeijer, W. Couzy and P.J. van der Houwen. A-stable parallel block methods for ordinary and integro-differential equations. Applied numerical mathematics, 9(3-5):267–281, 1992. https://doi.org/10.1016/0168-9274(92)90021-5
Y. Wei and Y. Chen. Legendre spectral collocation method for neutral and highorder Volterra integro-differential equation. Applied Numerical Mathematics, 81:15–29, 2014. https://doi.org/10.1016/j.apnum.2014.02.012
M.A. Wolfe and G.M. Phillips. Some methods for the solution of non-singular Volterra integro-differential equations. The Computer Journal, 11(3):334–336, 1968. https://doi.org/10.1093/comjnl/11.3.334
P.H.M. Wolkenfelt. The construction of reducible quadrature rules for Volterra integral and integro-differential equations. IMA Journal of Numerical Analysis, 2(2):131–152, 1982. https://doi.org/10.1093/imanum/2.2.131