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In this paper we propose a method for computing the Faddeeva function w(z):... erfc(-iz) via truncated modified trapezoidal rule approximations to integrals on the real line. Our starting point is the method due to Matta and Reichel (Math. Comp. 25 (1971), pp. 339-344) and Hunter and Regan (Math. Comp. 26 (1972), pp. 339-541). Addressing shortcomings flagged by Weideman (SIAM. J. Numer. Anal. 31 (1994), pp. 1497-1518), we construct approximations which we prove are exponentially convergent as a function of N+1, the number of quadrature points, obtaining error bounds which show that accuracies of 2x10-15 in the computation of w(z) throughout the complex plane are achieved with N=11, this confirmed by computations. These approximations, moreover, provably achieve small relative errors throughout the upper complex half-plane where w(z) is non-zero. Numerical tests suggest that this new method is competitive, in accuracy and computation times, with existing methods for computing w(z) for complex z [ABSTRACT FROM AUTHOR]

An implicit-explicit multistep method based on the backward difference formulae (BDF) is proposed for time discretization of parabolic equations with a non-self-adjoint operator. Implicit and explicit schemes are used for the self-adjoint and anti-self-adjoint parts of the operator, respectively. For a k-step method, some correction terms are added to the starting k - 1 steps to maintain kth-order convergence without imposing further compatibility conditions at the initial time. Long-time kth-order convergence for the numerical method is proved under the assumptions that the operator is coercive and that the non-self-adjoint part is low order. Such an operator often appears in practical computation (such as the Stokes--Darcy system) but may violate the standard sectorial angle condition used in the literature for analysis of BDF. In particular, the proposed method and analysis in this paper extend the long-time energy error analysis of the Stokes--Darcy system in Chen et al. [SIAM J. Numer. Anal., 51 (2013), pp. 2563--2584; Numer. Math., 134 (2016), pp. 857--879] to general symmetrized and decoupled BDF methods up to order 6 by using the generating function technique. [ABSTRACT FROM AUTHOR]