SIXTH ORDER EXPLICIT EXPONENTIAL ROSENBROCK-TYPE METHODS FOR SEMILINEAR PARABOLIC PROBLEMS

Yuhao Cong; Dongping Li

doi:10.11948/2013024

2013 Volume 3 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2013 > 3(4): 323-333

Next Article Previous Article

Yuhao Cong, Dongping Li. SIXTH ORDER EXPLICIT EXPONENTIAL ROSENBROCK-TYPE METHODS FOR SEMILINEAR PARABOLIC PROBLEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(4): 323-333. doi: 10.11948/2013024

Citation:

Yuhao Cong, Dongping Li. SIXTH ORDER EXPLICIT EXPONENTIAL ROSENBROCK-TYPE METHODS FOR SEMILINEAR PARABOLIC PROBLEMS[J]. Journal of Applied Analysis & Computation, 2013, 3(4): 323-333. doi: 10.11948/2013024

SIXTH ORDER EXPLICIT EXPONENTIAL ROSENBROCK-TYPE METHODS FOR SEMILINEAR PARABOLIC PROBLEMS

Yuhao Cong¹,
Dongping Li^1,2

1 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China;
2 Department of Mathematics, Changchun Normal University, Changchun, 130032, China

Fund Project:

Abstract

Abstract

The paper is concerned with the numerical analysis of high-order exponential Rosenbrock-type integrators for large-scale systems of stiff differential equations. The analysis is performed in a semigroup framework of semilinear evolution equations in Banach space. By expanding the errors of the numerical methods in terms of the solution, we further derive new order conditions and thus allows us to construct higher-order methods. A new and more general stiff error analysis is presented to show the converge results for variable step sizes.
- Exponential Rosenbrock-type methods /
- stiff order conditions /
- exponential integrators /
- variable step size
MSC: 65M12;65L06

FullText(HTML)

References

[1]	H. Berland, B. Owren and B. Skaflestad, B-series and order conditions for exponential integrators, SIAM J. Numer. Analy., 43(2005), 1715-1727. Google Scholar
[2]	J. C. Butcher, Numerical methods for ordinary differential equations, John Wiley and Sons Ltd, Chichester, 2008. Google Scholar
[3]	M. Caliari and A. Ostermann, Implementation of exponential Rosenbrocktype integrators, Appl. Numer. Math., 59(2009), 568C581. Google Scholar
[4]	K. J. Engel and R. Nagel, One-Parameter semigroups for linear evolution equations, Springer, New York, 2000. Google Scholar
[5]	M. Hochbruck and A. Ostermann, Exponential Runge-Kutta methods for parabolic problems, Appl. Numer. Math. 53(2005), 323-339. Google Scholar
[6]	M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilinear parabolic problems, SIAM J. Numer. Anal., 43(2006), 1069-1090. Google Scholar
[7]	M. Hochbruck, A. Ostermann and J. Schweitzer, Exponential Rosenbrock-Type methods, SIAM J. Numer. Anal., 47(2009), 786-803. Google Scholar
[8]	M. Hochbruck, C. Lubich and H. Selhofer, Exponential integrators for large systems of differential equations, SIAM J. Sci. Comput., 19(1998), 1552-1574. Google Scholar
[9]	M. Hochbruck and A. Ostermann, Exponential integrators, Acta Numerica, 19(2010), 209-286. Google Scholar
[10]	A. K. Kassam and L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM J. Sci. Comput., 26(2005), 1214C1233. Google Scholar
[11]	S. Krogstad, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys., 203(2005), 72C88. Google Scholar
[12]	V. T. Luan and A. Ostermann, Exponential Rosenbrock methods of order fiveconstruction, analysis and numerical comparisons, J. Comput. Appl. Math., 255(2014), 417-431. Google Scholar
[13]	J. Loffeld and M. Tokman, Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs, J. Comput. Appl. Math., 241(2013), 45-67. Google Scholar
[14]	M. Tokman, Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods, J. Comput. Phys., 213(2006), 748C776. Google Scholar