High order second derivative diagonally implicit multistage integration methods for ODEs
Abstract
Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.
Keyword : general linear methods, second derivative methods, order conditions, A− and L−stability, Fourier series
How to Cite
Sharifi, M., Abdi, A., Braś, M., & Hojjati, G. (2023). High order second derivative diagonally implicit multistage integration methods for ODEs. Mathematical Modelling and Analysis, 28(1), 53–70. https://doi.org/10.3846/mma.2023.16102
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Abdi and B. Behzad. Efficient Nordsieck second derivative general linear methods: construction and implementation. Calcolo, 55(28):1–16, 2018. https://doi.org/10.1007/s10092-018-0270-7
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
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A. Abdi and G. Hojjati. An extension of general linear methods. Numerical Algorithms, 57(2):149–167, 2011. https://doi.org/10.1007/s11075-010-9420-y
A. Abdi and G. Hojjati. Maximal order for second derivative general linear methods with Runge–Kutta stability. Applied Numerical Mathematics, 61(10):1046– 1058, 2011. https://doi.org/10.1016/j.apnum.2011.06.004
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B. Behzad, B. Ghazanfari and A. Abdi. Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs. Computational and Applied Mathematics, 37(4):5098–5112, 2018. https://doi.org/10.1007/s40314-018-0619-1
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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
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A.J. Christlieb, S. Gottlieb, Z. Grant and D.C. Seal. Explicit strong stability preserving multistage two-derivative time-stepping schemes. Journal of Scientific Computing, 68(3):914–942, 2016. https://doi.org/10.1007/s10915-016-0164-2
G. Dahlquist. A special stability problem for linear multistep methods. BIT Numerical Mathematics, 3(1):27–43, 1963. https://doi.org/10.1007/BF01963532
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
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 and G. Wanner. Solving ordinary differential equations II: Stiff and Differential-Algebraic Problems. Springer Berlin Heidelberg, 2010.
G. Hojjati, M.Y. Rahimi Ardabili and S.M. Hosseini. New second derivative multistep methods for stiff systems. Applied Mathematical Modelling, 30(5):466– 476, 2006. https://doi.org/10.1016/j.apm.2005.06.007
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: Numerical Methods for Solving Stiff Initial Value Problems. Proceeding, Oberwolfach 28.6.4.7.1981 (Dahlquist, G., Jeltsch, R., eds.). Bericht Nr. 9, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-5100 Aachen. Templergraben 55, D-5100 Aachen, 1981.
A.Y.J. Tsai, R.P.K. Chan and S. Wang. Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach. Numerical Algorithms, 65(3):687–703, 2014. https://doi.org/10.1007/s11075-014-9823-2
A. Abdi. Construction of high-order quadratically stable second-derivative general linear methods for the numerical integration of stiff ODEs. Journal of Computational and Applied Mathematics, 303:218–228, 2016. https://doi.org/10.1016/j.cam.2016.02.054
A. Abdi and B. Behzad. Efficient Nordsieck second derivative general linear methods: construction and implementation. Calcolo, 55(28):1–16, 2018. https://doi.org/10.1007/s10092-018-0270-7
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
A. Abdi and D. Conte. Implementation of second derivative general linear methods. Calcolo, 57(20):1–29, 2020. https://doi.org/10.1007/s10092-020-00370-w
A. Abdi and G. Hojjati. An extension of general linear methods. Numerical Algorithms, 57(2):149–167, 2011. https://doi.org/10.1007/s11075-010-9420-y
A. Abdi and G. Hojjati. Maximal order for second derivative general linear methods with Runge–Kutta stability. Applied Numerical Mathematics, 61(10):1046– 1058, 2011. https://doi.org/10.1016/j.apnum.2011.06.004
A. Abdi and G. Hojjati. Implementation of Nordsieck second derivative methods for stiff ODEs. Applied Numerical Mathematics, 94:241–253, 2015. https://doi.org/10.1016/j.apnum.2015.04.002
A. Baeza, S. Boscarino, P. Mulet, G. Russo and D. Zorío. Reprint of: Approximate Taylor methods for ODEs. Computers & Fluids, 169:87–97, 2018. https://doi.org/10.1016/j.compfluid.2018.03.058
B. Behzad, B. Ghazanfari and A. Abdi. Construction of the Nordsieck second derivative methods with RK stability for stiff ODEs. Computational and Applied Mathematics, 37(4):5098–5112, 2018. https://doi.org/10.1007/s40314-018-0619-1
J.C. Butcher. On the convergence of numerical solutions to ordinary differential equations. Mathematics of Computation, 20(93):1–10, 1966. https://doi.org/10.1090/S0025-5718-1966-0189251-X
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.C. Butcher and P. Sehnalová. Predictor–corrector Obreshkov pairs. Computing, 95(5):355–371, 2013. https://doi.org/10.1007/s00607-012-0258-0
A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Numerical Algorithms, 60(1):1–25, 2012. https://doi.org/10.1007/s11075-011-9509-y
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
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
Y.F. Chang and G. Corliss. ATOMFT: solving ODEs and DAEs using Taylor series. Computers & Mathematics with Applications, 28(10):209–233, 1994. https://doi.org/10.1016/0898-1221(94)00193-6
A.J. Christlieb, S. Gottlieb, Z. Grant and D.C. Seal. Explicit strong stability preserving multistage two-derivative time-stepping schemes. Journal of Scientific Computing, 68(3):914–942, 2016. https://doi.org/10.1007/s10915-016-0164-2
G. Dahlquist. A special stability problem for linear multistep methods. BIT Numerical Mathematics, 3(1):27–43, 1963. https://doi.org/10.1007/BF01963532
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
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 and G. Wanner. Solving ordinary differential equations II: Stiff and Differential-Algebraic Problems. Springer Berlin Heidelberg, 2010.
G. Hojjati, M.Y. Rahimi Ardabili and S.M. Hosseini. New second derivative multistep methods for stiff systems. Applied Mathematical Modelling, 30(5):466– 476, 2006. https://doi.org/10.1016/j.apm.2005.06.007
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: Numerical Methods for Solving Stiff Initial Value Problems. Proceeding, Oberwolfach 28.6.4.7.1981 (Dahlquist, G., Jeltsch, R., eds.). Bericht Nr. 9, Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-5100 Aachen. Templergraben 55, D-5100 Aachen, 1981.
A.Y.J. Tsai, R.P.K. Chan and S. Wang. Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach. Numerical Algorithms, 65(3):687–703, 2014. https://doi.org/10.1007/s11075-014-9823-2