Chen, D. J. L. The efficiency of singly-implicit Runge-Kutta methods for stiff differential equations. (English) Zbl 1291.65199 Numer. Algorithms 65, No. 3, 533-554 (2014). Summary: Singly-implicit Runge-Kutta methods are considered to be good candidates for stiff problems because of their good stability and high accuracy. The existing methods, SIRK (Singly-implicit Runge-Kutta), DESI (Diagonally Extendable Singly-implicit Runge-Kutta), ESIRK (Effective order Singly-implicit Rung-Kutta) and DESIRE (Diagonally Extended Singly-implicit Runge-Kutta Effective order) methods have been shown to be efficient for stiff differential equations, especially for high dimensional stiff problems. In this paper, we measure the efficiency for the family of singly-implicit Runge-Kutta methods using the local truncation error produced within one single step and the count of number of operations. Verification of the error and the computational costs for these methods using variable stepsize scheme are presented. We show how the numerical results are effected by the designed factors: additional diagonal-implicit stages and effective order. Cited in 3 Documents MSC: 65L04 Numerical methods for stiff equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations Keywords:singly-implicit Runge-Kutta methods; diagonally-implicit stages; effective order; stiff problems; stability; local truncation error; variable stepsize scheme; numerical results Software:DESIRE; RODAS PDFBibTeX XMLCite \textit{D. J. L. Chen}, Numer. Algorithms 65, No. 3, 533--554 (2014; Zbl 1291.65199) Full Text: DOI References: [1] Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff ODEs. SIAM J. Numer. Anal. 14, 1006-1021 (1977) · Zbl 0374.65038 [2] Burrage, K.: A special family of Runge-Kutta methods for solving stiff differential equations. BIT 18, 22-41 (1978) · Zbl 0384.65034 [3] Burrage, K., Butcher, J.C., Chipman, F.H.: STRIDE: Stable Runge-Kutta integrator for differential equations. Computational Mathematics Report No. 20. University of Auckland (1979) · Zbl 0908.65064 [4] Butcher, J.C.: The effective order of Runge-Kutta methods, conference on the numerical solution of differential equations. Lect. Notes Math. 109, 133-139 (1969) [5] Butcher, J.C.: On the implementation of implicit Runge-Kutta methods. BIT 16, 237-240 (1976) · Zbl 0336.65037 [6] Butcher, J.C.: The numerical analysis of ordinary differential equations. Wiley (2008) · Zbl 1167.65041 [7] Butcher, J.C., Cash, J.: Towards efficient Runge-Kutta methods for stiff systems. SIAM J. Numer. Anal. 27, 753-761 (1990) · Zbl 0702.65072 [8] Butcher, J.C., Chartier, P.: A generalization of Singly-Implicit Runge-Kutta methods. Appl. Numer. Math. 24, 343-350 (1997) · Zbl 0906.65076 [9] Butcher, J.C., Chartier, P.: The Effective Order of Singly-Implicit Runge-Kutta methods. Numer. Algo. 20, 269-284 (1999) · Zbl 0936.65089 [10] Butcher, J.C., Chen, D.J.L.: ESIRK methods and variable stepsize. Appl. Numer. Math. 28, 193-207 (1998) · Zbl 0927.65096 [11] Butcher, J.C., Chen, D.J.L.: On the implementation of ESIRK methods for stiff IVPs. Numer. Algo. 26, 201-218 (2001) · Zbl 0974.65075 [12] Chen, D.J.L.: The effective order of singly-implicit methods for stiff differential equations. PhD thesis, The University of Auckland, New Zealand (1998) · Zbl 0384.65034 [13] Butcher, J.C., Diamantakis, M.T.: DESIRE: diagonally extended singly implicit Runge-Kutta effective order methods. Numer. Algo. 17, 121-145 (1998) · Zbl 0908.65064 [14] Diamantakis, M.T.: Diagonally extended singly implicit Runge-Kutta methods for stiff initial value problems. PhD thesis, Imperial College, University of London (1995) [15] Hairer, E., Wanner, G.: Solving ordinary differential equations II, stiff and differential-algebraic problems. Springer-Verlag, Berlin (1987) · Zbl 1192.65097 [16] Kaps, P.: The Rosenbrock-type methods. In: Dalhquist, G., Jeltsch, R. (eds.) Numerical methods for stiff initial value problems (Proceeding, Oberwolfach). Bericht Nr. 9, Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Germany (1981) · Zbl 0469.65047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.