Traditional nonlinear elements of the electromagnetic transient simulation program (EMTP) use the numerical integration method, often employing the trapezoidal integration method as an implicit algorithm. However, when the number of nonlinear elements is high, the iterative format becomes computationally cumbersome. The semi-implicit format algorithms, based on the Rosenbrock real-time integration method, do not require iterations and can be applied directly in a recursive manner. However, as the number of nonlinear elements increases, it becomes necessary to invert the matrix that contains the Jacobian matrix, resulting in an increase of both algorithmic computation and inversion time with the dimensionality of the system. In this paper, a new numerical solution method is adopted, which combines the precise integration method. Specifically, the Calahan algorithm, with 2-stage and 3-order, is used to numerically integrate the Duhamel integration terms, and the accuracy of the Calahan algorithm is enhanced through Richardson extrapolation. The new algorithm can avoid the inverse of the matrix containing the Jacobian matrix, and the computational efficiency is equivalent to an explicit method, which greatly improves the computational efficiency of the Rosenbrock integration algorithm. The results verify the effectiveness and accuracy of the algorithm for typical power system nonlinear element electromagnetic transient simulation cases.