An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

The Legendre-based ultraspherical spectral method for ordinary differential equations (Olver, S. & Townsend, A. (2013) A fast and well-conditioned spectral method. SIAM Rev., 55, 462–489.) is combined with a formula for the convolution of two Legendre series (Hale, N. & Townsend, A. (2014a) An algorithm for the convolution of Legendre series. SIAM J. Sci. Comput., 36, A1207–A1220.) to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth, then the method is spectrally accurate and the resulting almost-banded linear systems can be solved with linear complexity.

Parallel Algorithm with Parameters Based on Alternating Direction for Solving Banded Linear Systems

An efficient parallel iterative method with parameters on distributed-memory multicomputer is investigated for solving the banded linear equations in this work. The parallel algorithm at each iterative step is executed using alternating direction by splitting the coefficient matrix and using parameters properly. Only it twice requires the communications of the algorithm between the adjacent processors, so this method has high parallel efficiency. Some convergence theorems for different coefficient matrices are given, such as a Hermite positive definite matrix or anM-matrix. Numerical experiments implemented on HP rx2600 cluster verify that our algorithm has the advantages over the multisplitting one of high efficiency and low memory space, which has a considerable advantage in CPU-times costs over the BSOR one. The efficiency for Example 1 is better than BSOR one significantly. As to Example 2, the acceleration rates and efficiency of our algorithm are better than the PEk inner iterative one.