Some efficient Nordsieck integration methods for IVPs

Iranian Journal of Numerical Analysis and Optimization

مقاله 6، دوره 8، شماره 1 - شماره پیاپی 13، خرداد 2018، صفحه 111-128   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v8i1.57845

نویسندگان

N. Barghi Oskouie؛ A. Abdi^* ؛ G. Hojjati

University of Tabriz, Tabriz,

چکیده

In this paper, in continuation of the construction of efficient numerical methods for stiff IVPs, we construct type two Nordsieck second derivative general linear methods with order p = s, where s is the number of internal stages, and stage order q = p. Implementation of the constructed methods with fixed and variable stepsize is discussed which verifies their efficiency.

کلیدواژه‌ها

Stiff differential equations؛ Nordsieck second derivative general linear methods؛ A- and L-stability؛ Variable stepsize implementation

مراجع

1. Abdi, A. Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs, J. Comput. Appl. Math. 303 (2016) 218–228.

2. Abdi, A., Bra´s, M., and Hojjati, G. On the construction of second derivative diagonally implicit multistage integration methods, Appl. Numer. Math. 76 (2014), 1–18.

3. Abdi, A. and Hojjati, G. High order second derivative methods with Runge–Kutta stability for the numerical solution of stiff ODEs, Iran. J. Numer. Anal. Optim. 5 (2015), 1–10.

4. Abdi, A. and Hojjati, G. Implementation of Nordsieck second derivative methods for stiff ODEs, Appl. Numer. Math. 94 (2015), 241–253.

5. Abdi, A. and Hojjati, G. An extension of general linear methods, Numer.

Algor. 57 (2011), no. 2, 149–167.

6. Abdi, A. and Hojjati, G. Maximal order for second derivative general linear methods with Runge–Kutta stability, Appl. Numer. Math. 61 (2011), no. 10, 1046–1058.

7. Barghi Oskouie, N., Hojjati, G., and Abdi. A. Efficient second derivative methods with extended stability regions for non-stiff IVPs. Comp. Appl. Math. DOI: 10.1007/s40314-017-0436-y.

8. Butcher, J. C. Numerical Methods for Ordinary Differential Equations, Wiley, New York, 2008.

9. Butcher, J. C. A transformation for the analysis of DIMSIMs. BIT 34(1994), no. 1, 25–32.

10. Butcher, J. C. and Chartier, P. Parallel general linear methods for stiff ordinary differential and differential algebraic equations, Appl. Numer. Math. 17 (1995), 213–222.

11. Butcher, J. C. and Hojjati, G. Second derivative methods with RK stability, Numer. Algor. 40 (2005), no. 4, 415–429.

12. Butcher, J. C. and Jackiewicz, Z. Construction of high order diagonally implicit multistage integration methods for ordinary differential equations. Appl. Numer. Math., 27 (1998), no. 1, 1–12.

13. Butcher, J. C. and Jackiewicz, Z. Implementation of diagonally implicit multistage integration methods for ordinary differential equations, SIAM J. Numer. Anal. 34 (1997), no. 6, 2119–2141.

14. Butcher, J. C. and Jackiewicz, Z. Diagonally implicit general linear methods for ordinary differential equations, BIT 33 (1993), 452–472.

15. Butcher, J. C. and Singh, A. D. The choice of parameters in parallel general linear methods for stiff problems, Appl. Numer. Math. 34 (2000), no. 1, 59–84.

16. Hairer, E. and Wanner, G. Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems, Springer, Berlin, 1996.

17. Jackiewicz, Z. General linear methods for ordinary differential equations, Wiley, New Jersey, 2009.

18. Mazzia, F. and Magherini, C. http://www.dm.uniba.it/ testset.

19. Wanner, G., Hairer, E., and Nørsett, S. P. Order stars and stability theorems, BIT, 18 (1978), no. 4, 475–489.