Diagonalizable Extended Backward Differentiation Formulas

We generalize the extended backward differentiation formulas (EBDFs) introduced by Cash and by Psihoyios and Cash so that the system matrix in the modified Newton process can be block-diagonalized, enabling an efficient parallel implementation. The purpose of this paper is to justify the use of diagonalizable EBDFs on parallel computers and to offer a starting point for the development of a variable stepsize-variable order method. We construct methods which are L-stable up to order p = 6 and which have the same computational complexity per processor as the conventional BDF methods. Numerical experiments with the order 6 method show that a speedup factor of between 2 and 4 on four processors can be expected.