Exponential Runge–Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge–Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method. For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This all...

This paper describes on-going research into the development of new computational algorithms and software which combine parallelism, space and time adaptivity, implicit stiff solvers, and multigrid methods. Such software is necessary in order to be able to attempt three-dimensional simulations of fundamental solidification phenomena at physically relevant parameter values. The research builds up...

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of N À 1 harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as N → ∞ and in the second case so that it approximates white noise. In both cases the fastest natural frequency of t...

Recently, an extension of peer two-step methods to parameter-dependent IVPs has been proposed by the author and E. Kostina where additional embedded stages allow cheap computation of solution derivatives (sensitivities) with respect to problem parameters. For each parameter only one additional ‘satellite’ stage was used which provides a highorder approximation to a neighboring solution. However...

Krylov implicit integration factor (IIF) methods were developed in Chen and Zhang (J Comput Phys 230:4336–4352, 2011) for solving stiff reaction–diffusion equations on high dimensional unstructured meshes. The methods were further extended to solve stiff advection–diffusion–reaction equations in Jiang and Zhang (J Comput Phys 253:368–388, 2013). Recently we studied the computational power of Kr...

K e y w o r d s R o s e n b r o c k methods, A-stable, Parallel algorithm, Stiff initial value problem, Adal> tivity. 1. I N T R O D U C T I O N We consider the numerical solution of systems of initial value ordinary differential equations (ODEs), i.e., initial value problems (IVPs), of the form y'(t) -f (y(t)) , y(to) = Yo, (1) where y : R --* R "~ and f : R "~ --* R m. Runge-Kutta methods app...

The parareal scheme (resp. PITA algorithm) proposed in [4] (resp. [1]) considers two levels of grids in time in order to split the domain in time-subdomains. A prediction of the solution is computed on the fine grid in parallel. Then at each end boundaries of time subdomains, the solution makes a jump with the previous initial boundary value (IBV) of the next time-subdomain . A correction of th...