A general four-parameter non-FSAL embedded Runge–Kutta algorithm of orders 6 and 4 in seven stages

• Published in: Applied mathematics and computation Vol. 143; no. 2; pp. 259 - 267
• Main Authors: El-Mikkawy, M.E.A, Eisa, M.M.M
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 10.11.2003; Elsevier
• Subjects: Embedded Runge–Kutta algorithm; Exact sciences and technology; FSAL; Initial value problems; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Order conditions; Ordinary differential equations; Sciences and techniques of general use; Order conditions; FSAL; Initial value problems; Embedded Runge–Kutta algorithm; Initial condition; 1993; Differential equation; Initial value problem; 1980; Embedding; Numerical method; Runge Kutta method; Vandermonde system; Algorithm; Exact solution; Vandermonde matrix
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(02)00358-2

The well-known Dormand–Prince embedded RK 5(4) 7FM algorithm [J. Comput. Appl. Math. 6 (1980) 19] is of the FSAL type and uses seven stages per step. This algorithm has been recommended by Shampine [Math. Comput. 46 (1986) 135] as a candidate for an efficient production RK code. In fact the new MATLAB function ode45 is based on this algorithm. Later on another efficient RK 5(4) 7 embedded algorithm, also having orders 5 and 4, is developed by Sharp and Smart [SIAM J. Sci. Comput. 14 (1993) 338]. The last algorithm uses seven stages per step and it is of the non-FSAL type. The current paper shows that by using seven stages per step a general four-parameter, non-FSAL embedded RK algorithm having orders 6 and 4 may be designed. A special algorithm, called RK 6(4) 7 new is obtained by using suitable choices for the free parameters. This new algorithm together with the RK 5(4) 7FM in [J. Comput. Appl. Math. 6 (1980) 19] and the RK 5(4) 7 in [SIAM J. Sci. Comput. 14 (1993) 338] are applied to some test problems, which have known exact solutions. It is found that the new algorithm is competitive comparing with the algorithms in [J. Comput. Appl. Math. 6 (1980) 19] and in [SIAM J. Sci. Comput. 14 (1993) 338].