Abstract
In this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra integral equations of Hammerstein type are constructed. The coefficients of the methods are expressed in terms of the values of the Laplace transform of the kernel. These methods have been suitably constructed in order to be implemented in an efficient way, thus leading to a very low computational cost both in time and in space. The order of convergence of the constructed methods is studied. The numerical experiments confirm the expected accuracy and computational cost.
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A. Bellen, Z. Jackiewicz, R. Vermiglio, and M. Zennaro, Stability analysis of Runge–Kutta methods for Volterra integral equations of second kind, IMA J. Numer. Anal., 10 (1990), pp. 103–118.
J. G. Blom and H. Brunner, The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 806–830.
H. Brunner and E. Messina, Time-stepping methods for Volterra–Fredholm integral equations, Rend. Mat. Appl., VII. Ser., 23 (2003), pp. 329–342.
H. Brunner and P. J. van der Houwen, The numerical solution of Volterra equations, CWI Monographs, vol. 3, North-Holland, Amsterdam, 1986.
A. Cardone, E. Messina, and E. Russo, A fast iterative method for Volterra–Fredholm integral equations, J. Comput. Appl. Math., 189(1–2) (2006), pp. 568–579.
D. Conte, I. Del Prete, Fast collocation methods for Volterra integral equations of convolution type, J. Comput. Appl. Math., 196(2) (2006), pp. 652–663.
M. R. Crisci, E. Russo, and A. Vecchio, On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation, BIT, 29 (1989), pp. 258–269.
F. de Hoog, R. Weiss, Implicit Runge–Kutta methods for second kind Volterra integral equations, Numer. Math., 23 (1975), pp. 199–213.
Z. S. Deligonoul, S. Bilgen, Solution of the Volterra equation of renewal theory with the Galerkin technique using cubic spplines, J. Stat. Comput. Simulation, 20 (1984), pp. 37–45.
D. Givoli, Numerical methods for problems in infinite domains, Elsevier Science Publishers, Amsterdam, 1992.
E. Hairer, C. Lubich, and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 532–541.
H. Han, L. Zhu, H. Brunner, and J. Ma, The numerical solution of parabolic Volterra integro-differential equations on unbounded spatial domains, Appl. Numer. Math., 55 (2005), pp. 83–99.
R. Hiptmair and A. Schädle, Non-reflecting boundary conditions for Maxwell’s equations, Computing, 71(3) (2003), pp. 265–292.
C. Lubich and A. Schädle, Fast convolution for non-reflecting boundary conditions, Siam. J. Sci. Comput., 24 (2002), pp. 161–182.
C. Lubich, Convolution quadrature and discretized operational calculus II, Numer. Math., 52 (1988), pp. 413–425.
R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, Menlo Park, CA, 1971.
M. Rizzardi, A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace Transform, ACM Trans. Math. Softw., 21(4) (1995), pp. 347–371.
P. W. Sharp and J. H. Verner: Some extended explicit Bel’tyukov pairs for Volterra integral equations of the second kind, SIAM J. Numer. Anal., 38(2) (2000), pp. 347–359.
A. Talbot, The accurate numerical inversion of Laplace Transforms, J. Inst. Math. Appl., 23 (1979), pp. 97–120.
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AMS subject classification (2000)
65R20, 45D05, 44A35, 44A10
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Capobianco, G., Conte, D., Del Prete, I. et al. Fast Runge–Kutta methods for nonlinear convolution systems of Volterra integral equations . Bit Numer Math 47, 259–275 (2007). https://doi.org/10.1007/s10543-007-0120-5
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DOI: https://doi.org/10.1007/s10543-007-0120-5