In this paper, a class of multistage and multivalue numerical methods called multiderivative extended Runge–Kutta–Nystrom (MERKN) methods are presented for the integration of multi-frequenc...

Multiderivative extended Runge–Kutta–Nyström methods for multi-frequency oscillatory systems

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.

Exponential almost Runge-Kutta methods for semilinear problems

We look at two families of Exponential Integrators, the multi-step Exponential Time Differencing Methods and the multi-stage Exponential Runge-Kutta Methods. A broader framework of Exponential General Linear Methods builds on these two families.

We extend the classical Almost Runge-Kutta Method family, of
multi-stage multi-value schemes, into an EI formulation, developing a new family of EARK. We extend the EGLM framework to include our new Exponential Almost Runge-Kutta Method. We call this new framework, Exponential Almost General Linear Method. This extension allows us to
develop new schemes which combine the features of the earlier families. We perform an analytical study of these schemes and verify that they demonstrate the necessary stability and convergence properties.

We then design a numerical, adaptive stepsize, differential equation solver. The core of the solver is powered by EIs, in particular our new family of EAGLMs. Experiments are performed against a comprehensive suite of test problems, with emphasis on stiff systems of semi-linear ODE's resulting from a spatial discretisation of PDE's.
These experiments reinforce existing results, showing that EIs significantly outperform existing solvers, such as Matlab's ODE15s. We demonstrate that our EAGLMs are the most efficient, and best performing, EIs.

The analysis and implementation of exponential almost Runge-Kutta methods for semilinear problems

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The analysis and implementation of exponential almost Runge-Kutta methods for semilinear problems

Exponential almost Runge-Kutta methods for semilinear problems