Abstract
We present a new class of one-step, multi-value Exponential Integrator (EI) methods referred to as Exponential Almost Runge-Kutta (EARK) methods which involve the derivatives of a nonlinear function of the solution. In order to approximate such derivatives to a sufficient accuracy, the EARK methods will be implemented within the broader framework of Exponential Almost General Linear Methods (EAGLMs) to accommodate past values of this nonlinear function and becoming multistep in nature as a consequence. Established EI methods, such as Exponential Time Differencing (ETD) methods, Exponential Runge-Kutta (ERK) methods and Exponential General Linear Methods (EGLMs) become special cases of EAGLMs. We present order conditions which facilitate the construction of two- and three-stage EARK methods and, when cast in an EAGLM format, we perform a stability analysis to enable a comparison with existing EI methods. We conclude with some numerical experiments which confirm the convergence order and also demonstrate the computational efficiency of these new methods.
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Communicated by Mechthild Thalhammer.
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Carroll, J., O’Callaghan, E. Exponential almost Runge-Kutta methods for semilinear problems. Bit Numer Math 53, 567–594 (2013). https://doi.org/10.1007/s10543-012-0416-y
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DOI: https://doi.org/10.1007/s10543-012-0416-y
Keywords
- Exponential integrators
- Exponential almost Runge-Kutta
- EARK
- Exponential almost general linear methods
- EAGLM
- Semilinear
- Stiff PDEs