Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T11:53:42.775Z Has data issue: false hasContentIssue false

Exponential Additive Runge-Kutta Methods for Semi-Linear Differential Equations

Published online by Cambridge University Press:  02 May 2017

Jingjun Zhao*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Teng Long
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Yang Xu*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
*
*Corresponding author. Email addresses:[email protected] (J. Zhao), [email protected] (Y. Xu)
*Corresponding author. Email addresses:[email protected] (J. Zhao), [email protected] (Y. Xu)
Get access

Abstract

Exponential additive Runge-Kutta methods for solving semi-linear equations are discussed. Related order conditions and stability properties for both explicit and implicit schemes are developed, according to the dimension of the coefficients in the linear terms. Several examples illustrate our theoretical results.

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Araújo, A.L., A note on B-stability of splitting methods, Comput. Vis. Sci. 6, 5357 (2004).Google Scholar
[2] Araújo, A.L., Murua, A. and Sanz-Serna, J.M., Symplectic methods based on decompositions, SIAM J. Numer. Anal. 34, 19261947 (1997).CrossRefGoogle Scholar
[3] Ascher, U.M., Ruuth, S.J. and Wetton, B.T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32, 797823 (1995).Google Scholar
[4] Caliari, M. and Ostermann, A., Implementation of exponential Rosenbrock-type integrators, Appl. Numer. Math. 59, 568581 (2009).Google Scholar
[5] Chou, C.S., Zhang, Y.T., Zhao, R. and Nie, Q., Numerical methods for stiff reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B 7, 515525 (2007).Google Scholar
[6] Christlieb, A., Morton, M., Ong, B. and Qiu, J.M., Semi-implicit integral deferred correction constructed with additive Runge-Kutta methods, Commun. Math. Sci. 9, 879902 (2011).Google Scholar
[7] Desoer, C. and Haneda, H., The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory, 19 (1972), pp. 480486.Google Scholar
[8] Dimitriu, G. and Stefănescu, R., Numerical experiments for reaction-diffusion equations using exponential integrators, Margenov, S., Vulkov, L.G. and Waśniewski, J. (Eds.): NAA 2008, LNCS 5434, pp. 249-256 (2009).Google Scholar
[9] Enright, W.H., Hull, T.E. and Lindberg, B., Comparing numerical methods for stiff systems of ODEs, BIT 15, 1048 (1975).Google Scholar
[10] Gondal, M.A., Exponential Rosenbrock integrators for option pricing, J. Comput. Appl. Math. 234, 11531160 (2010).Google Scholar
[11] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin Heidelberg (1981).Google Scholar
[12] Hochbruck, M. and Ostermann, A., Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal. 43, 10691090 (2005).Google Scholar
[13] Hochbruck, M. and Ostermann, A., Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math. 53, 323339 (2005).CrossRefGoogle Scholar
[14] Hochbruck, M., Ostermann, A. and Schweitzer, J., Exponential Rosenbrock-type methods, SIAM J. Numer. Anal. 47, 786803 (2009).Google Scholar
[15] Jiang, T. and Zhang, Y.T., Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations, J. Comput. Phys. 253, 368388 (2013).Google Scholar
[16] Kassam, A.K. and Trefethen, L.N., Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26, 12141233 (2005).Google Scholar
[17] Kennedy, C.A. and Carpenter, M.H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math. 44, 139181 (2003).Google Scholar
[18] Maset, S. and Zennaro, M., Unconditional stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations, Math. Comp. 78, 957967 (2009).Google Scholar
[19] Minchev, B. and Wright, W.M., A Review of Exponential Integrators for First Order Semi-linear Problems, Tech. report 2/05, Department of Mathematical Sciences, Norwegian University of Science and Technology (2005).Google Scholar
[20] Najm, H.N., Wyckoff, P.S. and Knio, O.M., A semi-implicit numerical scheme for reacting flow: I. stiff chemistry, J. Comput. Phys. 143, 381402 (1998).Google Scholar
[21] Ostermann, A. and Thalhammer, M., Positivity of Exponential Multistep Methods, Numerical Mathematics and Advanced Applications, Springer, Berlin, pp. 564571 (2006).Google Scholar
[22] Ostermann, A., Thalhammer, M. and Wright, W.M., A class of explicit exponential general linear methods, BIT 46, 409431 (2006).CrossRefGoogle Scholar
[23] Pareschi, L. and Russo, G., Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations, in Recent Trends in Numerical Analysis, Adv. Theory Comput. Math. 3, Nova Sci. Publ., Huntington, New York, pp. 269288 (2001).Google Scholar
[24] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).Google Scholar
[25] Verwer, J.G., S-stability properties for generalized Runge-Kutta methods, Numer. Math. 27, 359370 (1976).Google Scholar