Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control

SIAM Journal on Scientific Computing, 32(4) (2010), 1695-1723
http://dx.doi.org/10.1137/090764840

In this paper, we present variable-stepsize explicit parallel peer methods grounded in the interpolation idea. Approximation, stability, and convergence are studied in detail. In particular, we prove that some interpolating peer methods are stable on any variable mesh in practice. Double quasi consistency is utilized to introduce an efficient global error estimation formula in the numerical methods under discussion. The main advantage of these new adaptive schemes is the fact that the leading terms of their local and true errors coincide. Thus, controlling the local error of such methods by cheap standard techniques automatically regulates their global error as well. Numerical experiments of this paper support theoretical results presented below and illustrate how the new global error control concept works in practice. We also conduct a comparison with explicit ODE solvers in MATLAB.