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Based on a generalization of discrete Gronwall inequality and Navots quadrature rule, this paper presents a high accuracy quadrature method for solving system of weakly singular nonlinear Volterra integral equa tions of the second kind and integro-differential equations.An iterative algorithm for solving the discrete system is proposed, which possesses the high accuracy order O(h2+α).After the asymptotic expansion of errors is proved, we can obtain an approximation with a higher accuracy order using Richardson extrapolation.An a posteriori error estimation is pro vided.Some numerical results are presented to show the efficiency of our methods.