Convergence of discrete-time relaxation methods based on Volterra backward differentiation formulas (Q1568175)

The author proposes the discrete-time relaxation methods based on indirect backward differentiation methods for the numerical solution of large system of second kind Volterra integral equations (VIEs). A necessary and sufficient condition for the convergence of the iterative process in the case of a system of linear VIEs is obtained. Some more practical sufficient conditions not involving the eigenvalues of the convergence matrix are proved. The case of a linear convolution system is considered and a sufficient condition for the convergence of the waveform relaxation backward differentiation (WRBDF) method to the underlying method is furnished. This condition depends on the logarithmic norm of the chosen splitting and is independent of the stepsize and of the window. The case of parallel splittings is analyzed. A theorem for the convergence of the methods based on Jacobi splitting whose hypotheses require some condition on the kernel which are very easy to check is proved. A useful criterion for the choice of the Richardson splitting leading to convergent WRBDF method is obtained.