The analysis and implementation of exponential almost Runge-Kutta methods for semilinear problems
O'Callaghan, Eóin (2011) The analysis and implementation of exponential almost Runge-Kutta methods for semilinear problems. PhD thesis, Dublin City University.
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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.
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