Abstract
In this paper we describe the construction of second derivative general linear method with Runge–Kutta stability property preserving the strong stability properties of spatial discretizations. Then we present such methods that are obtained by the solution of the constrained minimization problem with nonlinear inequality constraints, corresponding to the strong stability preserving property of these methods, and equality constraints, corresponding to the order, stage order and Runge–Kutta stability conditions. The derived methods are of order \(p=q\) with \(r=2\) and \(s=p\) or \(s=p+1\), of order \(p=q=s=r-1\) and of order \(p=q+1=s=r\), where q, s and r are the stage order, the number of internal and the number of external stages, respectively. Efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in solving partial differential equations with smooth and discontinues initial data are shown by some numerical experiments.
Similar content being viewed by others
References
Abdi, A., Hojjati, G.: Maximal order for second derivative general linear methods with Runge–Kutta stability. Appl. Numer. Math. 61, 1046–1058 (2011)
Abdi, A., Behzad, B.: Efficient Nordsieck second derivative general linear methods: construction and implementation. Calcolo 55(28), 1–16 (2018)
Abdi, A., Braś, M., Hojjati, G.: On the construction of second derivative diagonally implicit multistage integration methods. Appl. Numer. Math. 76, 1–18 (2014)
Abdi, A., Hojjati, G.: Implementation of Nordsieck second derivative methods for stiff ODEs. Appl. Numer. Math. 94, 241–253 (2015)
Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Butcher, J.C., Hojjati, G.: Second derivative methods with RK stability. Numer. Algorithms 40, 415–429 (2005)
Califano, G., Izzo, G., Jackiewicz, Z.: Strong stability preserving general linear methods with Runge–Kutta stability. J. Sci. Comput. 76, 943–968 (2018)
Christlieb, A.J., Gottlieb, S., Grant, Z.J., Seal, D.C.: Explicit strong stability preserving multistage two-derivative time-stepping schemes. J. Sci. Comput. 68, 914–942 (2016)
Constantinescu, E.M., Sandu, A.: Optimal explicit strong-stability-preserving general linear methods. J. Sci. Comput. 32, 3130–3150 (2010)
Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 2743 (1963)
Ferracina, L., Spijker, M.N.: An extension and analysis of the Shu–Osher representation of Runge–Kutta methods. Math. Comput. 74, 201–219 (2004)
Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-boundedness in general Runge–Kutta procedures. Appl. Numer. Math. 53, 265–279 (2005)
Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)
Grant, Z., Gottlieb, S., Seal, D.C.: A strong stability preserving analysis for multistage two-derivative time-stepping schemes based on Taylor series conditions. Commun. Appl. Math. Comput. 1, 21–59 (2019)
Gottlieb, S.: On high order strong stability preserving Runge–Kutta methods and multistep time discretizations. J. Sci. Comput. 25, 105–127 (2005)
Gottlieb, S., Ketcheson, D.I., Shu, Chi-Wang: High order strong stability preserving time discretizations. J. Sci. Comput. 38, 251–289 (2009)
Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific, Hackensack (2011)
Gottlieb, S., Ruuth, S.J.: Optimal strong-stability-preserving time stepping schemes with fast downwind spatial discretizations. J. Sci. Comput. 27, 289–303 (2006)
Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
Hesthaven, J., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time Dependent Problems. Cambridge monographs of applied and computational mathematics. Cambridge University Press, Cambridge (2007)
Higueras, I.: On strong stability preserving time discretization methods. J. Sci. Comput. 21, 193–223 (2004)
Higueras, I.: Monotonicity for Runge–Kutta methods: inner product norms. J. Sci. Comput. 24, 97–117 (2005)
Higueras, I.: Representations of Runge–Kutta methods and strong stability preserving methods. SIAM J. Numer. Anal. 43, 924–948 (2005)
Hundsdorfer, W., Ruuth, S.J.: On monotonicity and boundedness properties of linear multistep methods. Math. Comput. 75, 655–672 (2005)
Hundsdorfer, W., Ruuth, S.J., Spiteri, R.J.: Monotonicity-preserving linear multistep methods. SIAM J. Numer. Anal. 41, 605–623 (2003)
Izzo, G., Jackiewicz, Z.: Strong stability preserving transformed DIMSIMs. J. Comput. Appl. Math. 343, 174–188 (2018)
Izzo, G., Jackiewicz, Z.: Strong stability preserving general linear methods. J. Sci. Comput. 65, 271–298 (2015)
Izzo, G., Jackiewicz, Z.: Strong stability preserving multistage integration methods. Math. Model. Anal. 20, 552–577 (2015)
Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)
Jackiewicz, Z., Tracogna, S.: A general class of two-step Runge–Kutta methods for ordinary differential equations. SIAM J. Numer. Anal. 32, 1390–1427 (1995)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Ketcheson, D.I., Gottlieb, S., Macdonald, C.B.: Strong stability preserving two-step Runge–Kutta methods. SIAM J. Numer. Anal. 49, 2618–2639 (2011)
Moradi, A., Farzi, J., Abdi, A.: Strong stability preserving second derivative general linear methods. J. Sci. Comput. 81, 392–435 (2019)
Moradi, A., Farzi, J., Abdi, A.: Order conditions for second derivative general linear methods. J. Comput. Appl. Math. (to appear)
Moradi, A., Abdi, A., Farzi, J.: Strong stability preserving diagonally implicit multistage integration methods. Appl. Numer. Math. 150, 536–558 (2020)
Ruuth, S.J., Hundsdorfer, W.: High-order linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 209, 226–248 (2005)
Seal, D.C., Guclu, Y., Christlieb, A.J.: High-order multiderivative time integrators for hyperbolic conservation laws. J. Sci. Comput. 60, 101–140 (2014)
Spijker, M.N.: Stepsize conditions for general monotonicity in numerical initial value problems. SIAM J. Numer. Anal. 45, 1226–1245 (2007)
Spiteri, R.J., Ruuth, S.J.: A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40, 469–491 (2002)
Shu, C.-W.: Total-variation diminishing time discretizations. J. Sci. Comput. 9, 1073–1084 (1988)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Tsai, A.Y.J., Chan, R.P.K., Wang, S.: Two-derivative Runge–Kutta methods for PDEs using a novel discretization approach. Numer. Algorithms 65, 687–703 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Moradi, A., Abdi, A. & Farzi, J. Strong Stability Preserving Second Derivative General Linear Methods with Runge–Kutta Stability. J Sci Comput 85, 1 (2020). https://doi.org/10.1007/s10915-020-01306-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-020-01306-w
Keywords
- General linear methods
- Second derivative methods
- Monotonicity
- Strong stability preserving
- Order conditions
- Runge–Kutta stability