*Abstract:* It is well known that classical Linear Multistep Formulas (LMFs) have some advantages with respect to Runge-Kutta methods such as the easier analysis of their properties and the same accuracy at each point (they have no internal stages to discard). On the other hand, they transform a first-order continuous problem in a kth-order discrete one. Such transformation has the undesired effect of introducing spurious solutions to be kept under control. It is such control which is responsible of the main drawbacks (e.g.,the two Dahlquist barriers) of LMFs with respect to Runge-Kutta methods. However, the control of the parasitic solutions is much easier if the continuous problem is transformed into a discrete boundary value problem.Starting from such idea, a new class of multistep methods, called Boundary Value Methods (BVMs), was proposed in the nineties [2]. The resulting schemes are free of order barriers; in particular, there exist A-stable BVMs of arbitrarily high order.In this talk, we shall describe the basic idea on which BVMs relies and discuss their accuracy and stability properties. In addition, a new family of convolution quadratures for the numerical solution of fractional differential equations will be presented. These schemes introduced in [1] are based on the BVM approach and overcome the barrier established in [3].

References:

[1] L. Aceto, C. Magherini, and P. Novati, Fractional convolution quadrature based on Generalized Adams Methods, Calcolo (2013), doi:10.1007/s10092-013-0094-4, in press.

[2] L. Brugnano and D. Trigiante, Solving ODEs by Linear Multistep Initial and Boundary Value Methods, Gordon & Breach, Amsterdam, 1998.

[3] C. Lubich, A stability analysis of convolution quadrature for Abel-Volterra integral equations, IMA J. Numer. Anal. 6 (1986) 87-101.