  We present a numerical integration scheme for evaluating the convolution of a
Green's function with a screened Coulomb potential on the real axis in the GW
approximation of the self energy. Our scheme takes the zero broadening limit in
Green's function first, replaces the numerator of the integrand with a
piecewise polynomial approximation, and performs principal value integration on
subintervals analytically. We give the error bound of our numerical integration
scheme and show by numerical examples that it is more reliable and accurate
than the standard quadrature rules such as the composite trapezoidal rule. We
also discuss the benefit of using different self energy expressions to perform
the numerical convolution at different frequencies.
