Wang, Chengshan; Fu, Xiaopeng; Li, Peng; Wu, Jianzhong Accurate dense output formula for exponential integrators using the scaling and squaring method. (English) Zbl 1315.65065 Appl. Math. Lett. 43, 101-107 (2015). Summary: This paper proposes an accurate dense output formula for exponential integrators. The computation of matrix exponential function is a vital step in implementing exponential integrators. By scrutinizing the computational process of matrix exponentials using the scaling and squaring method, valuable intermediate results in this process are identified and then used to establish a dense output formula. Efficient computation of dense outputs by the proposed formula enables time integration methods to set their simulation step sizes more flexibly. The efficacy of the proposed formula is verified through numerical examples from the power engineering field. MSC: 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65L04 Numerical methods for stiff equations Keywords:exponential integrator; dense output formula; matrix exponential; stiff equations; scaling and squaring method; numerical example Software:MATLAB expm PDFBibTeX XMLCite \textit{C. Wang} et al., Appl. Math. Lett. 43, 101--107 (2015; Zbl 1315.65065) Full Text: DOI References: [1] Hochbruck, M.; Ostermann, A., Exponential integrators, Acta Numer., 19, 209-286 (2010) · Zbl 1242.65109 [2] Al-Mohy, A. H.; Higham, N. J., Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33, 488-511 (2011) · Zbl 1234.65028 [3] Moler, C.; Van Loan, C., Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, 1, 3-49 (2003) · Zbl 1030.65029 [4] Higham, N. J., The scaling and squaring method for the matrix exponential revisited, SIAM Rev., 51, 4, 747-764 (2009) · Zbl 1178.65040 [5] Al-Mohy, A. H.; Higham, N. J., A new scaling and squaring algorithm for the matrix exponential, SIAM J. Matrix Anal. Appl., 31, 970-989 (2009) · Zbl 1194.15021 [6] Dommel, H., Digital computer solution of electromagnetic transient in single and multiphase networks, IEEE Trans. Power Appar. Syst., 88, 4, 388-399 (1969) [7] Hochbruck, M.; Ostermann, A.; Schweitzer, J., Exponential Rosenbrock-type methods, SIAM J. Numer. Anal., 47, 786-803 (2009) · Zbl 1193.65119 [8] Caliari, M.; Ostermann, A., Implementation of exponential Rosenbrock-type integrators, Appl. Numer. Math., 59, 568-581 (2009) · Zbl 1160.65318 [9] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29, 209-228 (1992) · Zbl 0749.65030 [10] Hochbruck, M.; Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, 1911-1925 (1997) · Zbl 0888.65032 [11] Brezinski, C.; Ieea, U.; Van Iseghem, J., A taste of Padé approximation, Acta Numer., 4, 53-103 (1995) · Zbl 0827.41010 [12] Kersting, W. H., Radial distribution test feeders, IEEE Trans. Power Syst., 6, 3, 975-985 (1991) 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.