Abstract
Exponential Runge–Kutta methods have shown to be competitive for the time integration of stiff semilinear parabolic PDEs. The current construction of stiffly accurate exponential Runge–Kutta methods, however, relies on a convergence result that requires weakening many of the order conditions, resulting in schemes whose stages must be implemented in a sequential way. In this work, after showing a stronger convergence result, we are able to derive two new families of fourth- and fifth-order exponential Runge–Kutta methods, which, in contrast to the existing methods, have multiple stages that are independent of one another and share the same format, thereby allowing them to be implemented in parallel or simultaneously, and making the methods to behave like using with much less stages. Moreover, all of their stages involve only one linear combination of the product of \(\varphi \)-functions (using the same argument) with vectors. Overall, these features make these new methods to be much more efficient to implement when compared to the existing methods of the same orders. Numerical experiments on a one-dimensional semilinear parabolic problem, a nonlinear Schrödinger equation, and a two-dimensional Gray–Scott model are given to confirm the accuracy and efficiency of the two newly constructed methods.
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References
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)
Berland, H., Skaflestad, B.: Solving the nonlinear Schrödinger equation using exponential integrators. Technical report (2005)
Berland, H., Skaflestad, B., Wright, W.M.: Expint–a matlab package for exponential integrators. ACM Trans. Math. Softw. 33(1), 4-es (2007)
Caliari, M., Kandolf, P., Ostermann, A., Rainer, S.: The Leja method revisited: backward error analysis for the matrix exponential. SIAM J. Sci. Comp. 38(3), A1639–A1661 (2016)
Cazenave, T.: An introduction to nonlinear Schrödinger equations, vol. 22. Universidade Federal do Rio de Janeiro, Centro de Ciências Matemáticas e da (1989)
Gaudreault, S., Pudykiewicz, J.: An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere. J. Comput. Phys. 322, 827–848 (2016)
Gray, P., Scott, S.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system \(A+ 2B \rightarrow 3B; B\rightarrow C\). Chem. Eng. Sci. 39(6), 1087–1097 (1984)
Hochbruck, M., Ostermann, A.: Explicit exponential Runge-Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2005)
Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)
Ju, L., Wang, Z.: Exponential time differencing Gauge method for incompressible viscous flows. Commun. Comput. Phys. 22(2), 517–541 (2017)
Luan, V.T.: High-order exponential integrators. Ph.D. thesis, University of Innsbruck (2014)
Luan, V.T.: Fourth-order two-stage explicit exponential integrators for time-dependent PDEs. Appl. Numer. Math. 112, 91–103 (2017)
Luan, V.T., Michels, D.: Efficient exponential time integration for simulating nonlinear coupled oscillators, J. Compt. Appl. Math. (Revised) (2020)
Luan, V.T., Ostermann, A.: Exponential B-series: the stiff case. SIAM J. Numer. Anal. 51, 3431–3445 (2013)
Luan, V.T., Ostermann, A.: Explicit exponential Runge-Kutta methods of high order for parabolic problems. J. Comput. Appl. Math. 256, 168–179 (2014)
Luan, V.T., Ostermann, A.: Exponential Rosenbrock methods of order five-construction, analysis and numerical comparisons. J. Comput. Appl. Math. 255, 417–431 (2014)
Luan, V.T., Ostermann, A.: Stiff order conditions for exponential Runge–Kutta methods of order five. In: H.B. et al. (ed.) Modeling, Simulation and Optimization of Complex Processes-HPSC 2012, pp. 133–143. Springer, Berlin (2014)
Luan, V.T., Ostermann, A.: Parallel exponential Rosenbrock methods. Comput. Math. Appl. 71, 1137–1150 (2016)
Luan, V.T., Pudykiewicz, J.A., Reynolds, D.R.: Further development of efficient and accurate time integration schemes for meteorological models. J. Comput. Phys. 376, 817–837 (2019)
Michels, D.L., Luan, V.T., Tokman, M.: A stiffly accurate integrator for elastodynamic problems. ACM Trans. Graph. 36(4), 116 (2017)
Niesen, J., Wright, W.M.: Algorithm 919: A Krylov subspace algorithm for evaluating the \(\varphi \)-functions appearing in exponential integrators. ACM Trans. Math. Soft. 38(3), 22 (2012)
Pieper, K., Sockwell, K.C., Gunzburger, M.: Exponential time differencing for mimetic multilayer ocean models. J. Comput. Phys. 398, 817–837 (2019)
Acknowledgements
The author would like to thank Reviewer 1 for the valuable comments and helpful suggestions. He would like also to thank the National Science Foundation, which supported this research under award NSF DMS–2012022. He also thanks Mississippi State University’s Center for Computational Science (CCS) for providing some resources at the High Performance Computing Collaboratory (HPCC).
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Communicated by Mechthild Thalhammer.
Dedicated to Professor Alexander Ostermann on the occasion of his 60th birthday.
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This work has been supported in part by National Science Foundation through award NSF DMS–2012022.
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Luan, V.T. Efficient exponential Runge–Kutta methods of high order: construction and implementation. Bit Numer Math 61, 535–560 (2021). https://doi.org/10.1007/s10543-020-00834-z
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DOI: https://doi.org/10.1007/s10543-020-00834-z