# Numerical solution methods for implicit Runge-Kutta methods of arbitrarily high order

## Main Article Content

## Abstract

In this study we consider an efficient implementation of Implicit Runge-Kutta methods for solving large systems of ordinary differential equations that originate from finite element discretization of the heat and similar equations, to be solved on large time intervals. The main contribution of this work is to show how to implement a fully stage-parallel version of the method, utilizing the dominance of the block lower triangular part of the quadrature matrix, and to illustrate it numerically. Its usage for the solution of algebraic-differential equations is also touched.

## Article Details

How to Cite

Axelsson, O., & Neytcheva, M.
(2020).
Numerical solution methods for implicit Runge-Kutta methods of arbitrarily high order.

*Proceedings Of The Conference Algoritmy,*, 11 - 20. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1546/810
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## References

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[10] O. Axelsson, R. Blaheta, and R. Kohut, Preconditioning methods for high-order strongly stable time integration methods with an application for a DAE problem, Numer. Linear Algebra Appl., 22 (2015), pp. 930–949.

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[13] O. Axelsson, On the efficiency of a class of A-stable methods, Nordisk Tidskr. Informationsbehandling (BIT), 14 (1974), pp. 279–287.

[14] P. J. van der Houwen and B. P. Sommeijer, Analysis of parallel diagonally implicit iteration of Runge-Kutta methods, Parallel methods for ordinary differential equations (Grado, 1991). Appl. Numer. Math. 11 (1993), pp. 169–188.

[15] J. C. Butcher, Diagonally-implicit multi-stage integration methods, Appl. Numer. Math, 11 (1993), pp. 347–363.

[16] Z. Zlatev, Modified diagonally implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 321–334.

[17] T. J. R. Hughes and G. M. Hulbert, Space-time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Math., 66 (1988), pp. 339–363.

[18] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43–77.

[19] O. Steinbach and H. Yang, Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D, Numer. Linear Algebra Appl., 25 (2018), e2143.

[20] J. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc., 42 (1937), pp. 461–496.

[21] O. Axelsson, D. Lukáš, Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems, J. Numer. Math., 27 (2019), pp. 1–21.

[22] H. Chen, Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations, Appl. Math. Modelling, 40 (2016), pp. 4429–4440.

[23] E. Hairer, C. Lubich and M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT, 28 (1988), pp. 678–700.

[24] O. Axelsson, R. Blaheta and T. Luber, Preconditioners for mixed FEM solution of stationary and nonstationary porous media flow problems, LSSC 2015, I. Lirkov et al (Eds.), LNCS 9374 (2015), pp. 3–14.

[25] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003.

[26] Y. Notay, An aggregation-based algebraic multigrid method, ETNA, 37 (2010), pp. 123–146.

[2] G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), pp. 27–43.

[3] O. Axelsson, A class of A-stable methods, Nordisk Tidskr. Informationsbehandling (BIT), 9 (1969), pp. 185–199.

[4] B. L. Ehle, High order A-stable methods for the numerical solution of D.E.s, BIT, 8 (1968), pp. 276–278.

[5] J. C. Butcher, Implicit Runge-Kutta processes, Math. Comput., 18 (1964), pp. 50–64.

[6] O. Axelsson, Global integration of differential equations through Lobatto quadrature, Nordisk Tidskr. Informationsbehandling, 4 (1964), pp. 69–86.

[7] B. L. Ehle, A-stable methods and Padé approximations to the exponential, SIAM J. Math. Anal., 4 (1973), pp. 671–680.

[8] J. D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, New York, 1992.

[9] L. Petzold, Order results for implicit Runge-Kutta methods, applied to differential-algebraic systems, SIAM J Numer Anal., 23 (1986), pp. 837–852.

[10] O. Axelsson, R. Blaheta, and R. Kohut, Preconditioning methods for high-order strongly stable time integration methods with an application for a DAE problem, Numer. Linear Algebra Appl., 22 (2015), pp. 930–949.

[11] E. Hairer and G. Wanner, Algebraically stable and implementable Runge-Kutta methods of higher order, SIAM J. Numer. Anal., 18 (1981), pp. 1098–1108.

[12] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, 1991.

[13] O. Axelsson, On the efficiency of a class of A-stable methods, Nordisk Tidskr. Informationsbehandling (BIT), 14 (1974), pp. 279–287.

[14] P. J. van der Houwen and B. P. Sommeijer, Analysis of parallel diagonally implicit iteration of Runge-Kutta methods, Parallel methods for ordinary differential equations (Grado, 1991). Appl. Numer. Math. 11 (1993), pp. 169–188.

[15] J. C. Butcher, Diagonally-implicit multi-stage integration methods, Appl. Numer. Math, 11 (1993), pp. 347–363.

[16] Z. Zlatev, Modified diagonally implicit Runge-Kutta methods, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 321–334.

[17] T. J. R. Hughes and G. M. Hulbert, Space-time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Math., 66 (1988), pp. 339–363.

[18] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43–77.

[19] O. Steinbach and H. Yang, Comparison of algebraic multigrid methods for an adaptive space-time finite-element discretization of the heat equation in 3D and 4D, Numer. Linear Algebra Appl., 25 (2018), e2143.

[20] J. Shohat, On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc., 42 (1937), pp. 461–496.

[21] O. Axelsson, D. Lukáš, Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems, J. Numer. Math., 27 (2019), pp. 1–21.

[22] H. Chen, Kronecker product splitting preconditioners for implicit Runge-Kutta discretizations of viscous wave equations, Appl. Math. Modelling, 40 (2016), pp. 4429–4440.

[23] E. Hairer, C. Lubich and M. Roche, Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations, BIT, 28 (1988), pp. 678–700.

[24] O. Axelsson, R. Blaheta and T. Luber, Preconditioners for mixed FEM solution of stationary and nonstationary porous media flow problems, LSSC 2015, I. Lirkov et al (Eds.), LNCS 9374 (2015), pp. 3–14.

[25] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, 2003.

[26] Y. Notay, An aggregation-based algebraic multigrid method, ETNA, 37 (2010), pp. 123–146.