Parallel iteration of the extended backward differentiation formulas (Q2713133)

The subject of this paper is the proposition of a parallelized algorithm realizing the so called extended backward differentiation formulas (EBDF) and the modified extended backward differentiation formulas (MEBDF). For the initial value problem for ordinary differential equations of the form NEWLINE\[NEWLINE{dy\over{dt}}=f(y)\;\;\;t\geq t_0, \tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEy(t_0)=y_0,NEWLINE\]NEWLINE the EBDF formulas by \textit{J. R. Cash} [Numer. Math. 34, 235-246 (1980; Zbl 0411.65040)] for evaluation of \(y_{n+1}\) are of the following form NEWLINE\[NEWLINEy_{n+1}=\sum_{j=1}^ka_jy_{n-j+1}+h[b_0f(y_{n+1})+b_1f(y_{n+2})]. \tag{2}NEWLINE\]NEWLINE In order to compute approximately the starting vector \(u_{n+1}\) for \(y_{n+1}\) and the vector \(y_{n+2}\), the standard implicit BDF corrector (playing here the role of a predictor) has to be used twice: NEWLINE\[NEWLINEu_{n+1}=\sum_{j=1}^k{\bar a}_jy_{n-j+1}+h{\bar b}_0f(u_{n+1}),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_{n+2}=\sum_{j=1}^k{\bar a}_jy_{n-j+2}+h{\bar b}_0f(u_{n+2}).NEWLINE\]NEWLINE The modified formulas differ a little. Since the method is implicit, the iteration is needed to compute vectors \(y_{n+1}\), \(u_{n+1}\) and \(u_{n+2}\). Parallelization enters at the level of resolution of the resulting system of the, in general, nonlinear equations. The iterative process is organized so that vectors \(u_{n+1}\), \(u_{n+2}\), \(y_{n+1}\) are computed simultaneously. The rate of convergence of the iterative process is discussed. The paper contains results of numerical experiments. Also timing of various versions of algorithms are compared.