A class of explicit second derivative general linear methods for non-stiff ODEs
Abstract
In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.
Keyword : non-stiff ODEs, general linear methods, second derivative methods, order conditions, quadratic stability
How to Cite
Sharifi, M., Abdi, A., Braś, M., & Hojjati, G. (2024). A class of explicit second derivative general linear methods for non-stiff ODEs. Mathematical Modelling and Analysis, 29(4), 621–640. https://doi.org/10.3846/mma.2024.19325
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Movahedinejad, G. Hojjati and A. Abdi. Construction of Nordsieck second derivative general linear methods with inherent quadratic stability. Mathematical Modelling and Analysis, 22(1):60–77, 2017. https://doi.org/10.3846/13926292.2017.1269024
N. Barghi Oskoui, 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
J. Schütz, D.C. Seal and J. Zeifang. Parallel–in–time high–order multiderivative IMEX solvers. Journal of Scientific Computing, 90(1):1–36, 2022. https://doi.org/10.1007/s10915-021-01733-3
D.C. Seal, Y. Güçlü and A.J. Christlieb. High–order multiderivative time integrators for hyperbolic conservation laws. Journal of Scientific Computing, 60(1):101–140, 2014. https://doi.org/10.1007/s10915-013-9787-8
M. Sharifi, A. Abdi, M. Braś and G. Hojjati. High order second derivative diagonally implicit multistage integration methods for ODEs. Mathematical Modelling and Analysis, 28(1):53–70, 2023. https://doi.org/10.3846/mma.2023.16102
A.Y.J. Tsai, R.P.K. Chan and S. Wang. Two-derivative Runge–Kutta methods for PDEs using a novel iscretization approach. Numerical Algorithms, 65(3):687– 703, 2014. https://doi.org/10.1007/s11075-014-9823-2
M.Ö. Turaci and T. Öziş. Derivation of three–derivative Runge–Kutta methods. Numerical Algorithms, 74(1):247–265, 2017. https://doi.org/10.1007/s11075-016-0147-2
A. Abdi, M. Braś and G. Hojjati. On the construction of second derivative diagonally implicit multistage integration methods for ODEs. Applied Numerical Mathematics, 76:1–18, 2014. https://doi.org/10.1016/j.apnum.2013.08.006
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. 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
J.C. Butcher. On the convergence of numerical solutions to ordinary differential equations. Mathematics of Computation, 20(93):1–10, 1966. https://doi.org/10.2307/2004263
J.C. Butcher. Numerical methods for ordinary differential equations. John Wiley & Sons, 2016. https://doi.org/10.1002/9781119121534
J.C. Butcher and G. Hojjati. Second derivative methods with RK stability. Numerical Algorithms, 40(4):415–429, 2005. https://doi.org/10.1007/s11075-005-0413-1
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.R. Cash. Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM Journal on Numerical Analysis, 18(1):21–36, 1981. https://doi.org/10.1137/0718003
P.C. Chakravarti and M.S. Kamel. Stiffly stable second derivative multistep methods with higher order and improved stability regions. BIT Numerical Mathematics, 23(1):75–83, 1983. https://doi.org/10.1007/BF01937327
R.P.K. Chan and A.Y.J. Tsai. On explicit two-derivative Runge–Kutta methods. Numerical Algorithms, 53(2):171–194, 2010. https://doi.org/10.1007/s11075-009-9349-1
W.H. Enright. Second derivative multistep methods for stiff ordinary differential equations. SIAM Journal on Numerical Analysis, 11(2):321–331, 1974. https://doi.org/10.1137/0711029
A.K. Ezzeddine, G. Hojjati and A. Abdi. Sequential second derivative general linear methods for stiff systems. Bulletin of the Iranian Mathematical Society, 40(1):83–100, 2014.
Y. Fang, X. You and Q. Ming. Trigonometrically fitted two-derivative Runge–Kutta methods for solving oscillatory differential equations. Numerical Algorithms, 65(3):651–667, 2014. https://doi.org/10.1007/s11075-013-9802-z
S. Gottlieb, Z.J. Grant, J. Hu and R. Shu. High order strong stability preserving multiderivative implicit and IMEX Runge–Kutta methods with asymptotic preserving properties. SIAM Journal on Numerical Analysis, 60(1):423–449, 2022. https://doi.org/10.1137/21M1403175
G.K. Gupta. Implementing second-derivative multistep methods using the Nordsieck polynomial representation. Mathematics of Computation, 32(141):13–18, 1978. https://doi.org/10.1090/S0025-5718-1978-0478630-7
E. Hairer, S.P. Nørsett and G. Wanner. Solving ordinary differential equations I: nonstiff problems. Springer, Berlin Heidelberg, 2010. https://doi.org/10.1007/978-3-540-78862-1
E. Hairer and G. Wanner. Solving ordinary differential equations II: stiff and Differential–Algebraic Problems. Springer, Berlin Heidelberg, 2010.
Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, 2009. https://doi.org/10.1002/9780470522165
P. Kaps. Rosenbrock–type methods. In G. Dahlquist and R. Jeltsch(Eds.), Numerical Methods for Solving Stiff Initial Value Problems, volume 9 of Oberwolfach 28.6.4.7.1981 Bericht Nr. 9, Templergraben 55, D-5100 Aachen, 1981. Institut für Geometrie und Praktische Mathematik, RWTH Aachen.
A. Movahedinejad, G. Hojjati and A. Abdi. Construction of Nordsieck second derivative general linear methods with inherent quadratic stability. Mathematical Modelling and Analysis, 22(1):60–77, 2017. https://doi.org/10.3846/13926292.2017.1269024
N. Barghi Oskoui, 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
J. Schütz, D.C. Seal and J. Zeifang. Parallel–in–time high–order multiderivative IMEX solvers. Journal of Scientific Computing, 90(1):1–36, 2022. https://doi.org/10.1007/s10915-021-01733-3
D.C. Seal, Y. Güçlü and A.J. Christlieb. High–order multiderivative time integrators for hyperbolic conservation laws. Journal of Scientific Computing, 60(1):101–140, 2014. https://doi.org/10.1007/s10915-013-9787-8
M. Sharifi, A. Abdi, M. Braś and G. Hojjati. High order second derivative diagonally implicit multistage integration methods for ODEs. Mathematical Modelling and Analysis, 28(1):53–70, 2023. https://doi.org/10.3846/mma.2023.16102
A.Y.J. Tsai, R.P.K. Chan and S. Wang. Two-derivative Runge–Kutta methods for PDEs using a novel iscretization approach. Numerical Algorithms, 65(3):687– 703, 2014. https://doi.org/10.1007/s11075-014-9823-2
M.Ö. Turaci and T. Öziş. Derivation of three–derivative Runge–Kutta methods. Numerical Algorithms, 74(1):247–265, 2017. https://doi.org/10.1007/s11075-016-0147-2