Extrapolated Implicit–explicit Runge–kutta Methods

Abstract

We investigate a new class of implicit–explicit singly diagonally implicit Runge–Kutta methods for ordinary differential equations with both non-stiff and stiff components. The approach is based on extrapolation of the stage values at the current step by stage values in the previous step. This approach was first proposed by the authors in context of implicit–explicit general linear methods.

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

“…The first equation in (41) requires, at each time step, the solution of an m-dimensional nonlinear system in the unknowns {U ni } m i=1 . For every choice of the collocation parameters c 1 , .…”

“…An extensive analysis of the stability properties on basic test equations is contained in [14]. A possible future development may regard new multistep methods with some relaxing order conditions, which leave some parameters free to perform a numerical search for the methods with optimal stability properties, as done in [22,23,[41][42][43] in the context of ODEs.…”

Order conditions for general linear methods

“…Such high order and stage order methods with some desirable stability properties (large regions of absolute stability for explicit methods, A-, L-, and algebraic stability for implicit methods) were investigated in a series of papers [29,56,30,[16][17][18][19][20][21][22][23][24][25][26][27]33,34,[57][58][59] and the monograph [3]. In the next section we will derive the general order conditions for GLMs (1.5) without restrictions on stage order (except stage preconsistency and stage consistency conditions (3.11) and (3.12)) using the approach proposed by Albrecht [60][61][62][63][64] in the context of RK, composite and linear cyclic methods, and generalized by Jackiewicz and Tracogna [39,40] and Tracogna [43], to TSRK methods for ODEs, by Jackiewicz and Vermiglio [65] to general linear methods with external stages of different orders, and by Garrappa [66] to some classes of Runge-Kutta methods for Volterra integral equations with weakly singular kernels.…”

Super-convergent implicit–explicit Peer methods with variable step sizes

“…Another approach to construct such methods is the use of extrapolation as proposed by Schneider, Lang, and Hundsdorfer [16]. This idea goes back to Crouzeix [7] and was also used by Cardone, Jackiewicz, Sandu, and Zhang [5,6] and later on by Braś, Izzo, and Jackiewicz [2] to construct implicit-explicit general linear and Runge-Kutta methods. The procedure can be easily extended to variable step sizes for IMEX-Peer methods.…”