Exponentially fitted singly diagonally implicit Runge-Kutta methods

It is the purpose of this paper to derive diagonally implicit exponentially fitted (EF) Runge-Kutta methods for the numerical solution of initial value problems based on first order ordinary differential equations, whose solutions are supposed to exhibit an exponential behaviour. In addition to the standard approach for the derivation of EF methods, we provide a revised constructive technique that takes into account the contribution to the error inherited from the computation of the internal stages. The derived methods are then compared to those obtained by neglecting the contribution of the error associated to the internal stages, as classically done in the standard derivation of multistage EF-based methods. Standard and revised EF methods are then compared in terms of linear stability and numerical performances.

[1]  T. E. Simos,et al.  An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions , 1998 .

[2]  H. De Meyer,et al.  Frequency evaluation in exponential fitting multistep algorithms for ODEs , 2002 .

[3]  D. Hollevoet,et al.  The optimal exponentially-fitted Numerov method for solving two-point boundary value problems , 2009 .

[4]  Jesús Vigo-Aguiar,et al.  Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods , 2005, Math. Comput. Model..

[5]  Raffaele D'Ambrosio,et al.  Construction of the ef-based Runge-Kutta methods revisited , 2011, Comput. Phys. Commun..

[6]  Beatrice Paternoster,et al.  Two-step hybrid collocation methods for y"=f(x, y) , 2009, Appl. Math. Lett..

[7]  Jesús Vigo-Aguiar,et al.  Exponential fitted Gauss, Radau and Lobatto methods of low order , 2008, Numerical Algorithms.

[8]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[9]  Guido Vanden Berghe,et al.  Symplectic exponentially-fitted four-stage Runge–Kutta methods of the Gauss type , 2011, Numerical Algorithms.

[10]  J. M. Franco.  Exponentially fitted explicit Runge-Kutta-Nyström methods , 2004 .

[11]  Raffaele D'Ambrosio,et al.  Exponentially fitted two-step hybrid methods for y″=f(x, y) , 2011, J. Comput. Appl. Math..

[12]  Beatrice Paternoster,et al.  Two Step Runge-Kutta-Nyström Methods for Oscillatory Problems Based on Mixed Polynomials , 2003, International Conference on Computational Science.

[13]  Beatrice Paternoster,et al.  Two-step modified collocation methods with structured coefficient matrices , 2012 .

[14]  Raffaele D'Ambrosio,et al.  Numerical search for algebraically stable two-step almost collocation methods , 2013, J. Comput. Appl. Math..

[15]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[16]  Zacharias A. Anastassi,et al.  A Family of Exponentially-fitted Runge–Kutta Methods with Exponential Order Up to Three for the Numerical Solution of the Schrödinger Equation , 2007 .

[17]  Liviu Gr. Ixaru,et al.  P-stability and exponential-fitting methods for y″″ = f(x, y) , 1996 .

[18]  Beatrice Paternoster,et al.  Two-step almost collocation methods for ordinary differential equations , 2009, Numerical Algorithms.

[19]  H. De Meyer,et al.  Exponentially-fitted explicit Runge–Kutta methods , 1999 .

[20]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[21]  Beatrice Paternoster,et al.  Runge-Kutta(-Nystro¨m) methods for ODEs with periodic solutions based on trigonometric polynomials , 1998 .

[22]  R. D'Ambrosio,et al.  Parameter estimation in exponentially fitted hybrid methods for second order differential problems , 2011, Journal of Mathematical Chemistry.

[23]  John P. Coleman,et al.  Mixed collocation methods for y ′′ =f x,y , 2000 .

[24]  T. E. Simos,et al.  Exponentially fitted symplectic integrator. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Manuel Calvo,et al.  Sixth-order symmetric and symplectic exponentially fitted Runge-Kutta methods of the Gauss type , 2009 .

[26]  J. M. Franco.  Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators , 2002 .

[27]  D. Conte,et al.  Two-step diagonally-implicit collocation based methods for Volterra Integral Equations , 2012 .

[28]  Liviu Gr. Ixaru.  Runge-Kutta Methods with Equation Dependent Coefficients , 2012, NAA.

[29]  Beatrice Paternoster,et al.  Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations , 2011, Math. Comput. Simul..

[30]  Beatrice Paternoster,et al.  Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday , 2012, Comput. Phys. Commun..

[31]  Raffaele D'Ambrosio,et al.  Continuous two-step Runge–Kutta methods for ordinary differential equations , 2010, Numerical Algorithms.

[32]  Raffaele D'Ambrosio,et al.  Construction and implementation of highly stable two-step continuous methods for stiff differential systems , 2011, Math. Comput. Simul..

[33]  Beatrice Paternoster,et al.  Two Step Runge-Kutta-Nyström Methods for y'' = f(x, y) and P-Stability , 2002, International Conference on Computational Science.

[34]  Liviu Gr. Ixaru.  Runge-Kutta method with equation dependent coefficients , 2012, Comput. Phys. Commun..

[35]  Beatrice Paternoster,et al.  General linear methods for y′′ = f (y (t)) , 2012, Numerical Algorithms.

[36]  G. Vanden Berghe,et al.  Exponential fitted Runge--Kutta methods of collocation type: fixed or variable knot points? , 2003 .

[37]  Liviu Gr. Ixaru,et al.  Truncation Errors in Exponential Fitting for Oscillatory Problems , 2006, SIAM J. Numer. Anal..

[38]  H. De Meyer,et al.  Exponentially fitted Runge-Kutta methods , 2000 .

[39]  Beatrice Paternoster,et al.  Exponentially fitted two-step Runge-Kutta methods: Construction and parameter selection , 2012, Appl. Math. Comput..