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A multi-stage diagonally-implicit Runge-Kutta (DIRK) algorithm is applied to discretize the time variable in transient magnetic field computation using finite element method (FEM). A formulation, which has the same format as the backward Euler (BE) algorithm for both linear and nonlinear problems, is deduced for simple and ready numerical implementation. The DIRK algorithm is compared with the BE algorithm which is an effective and popular algorithm in FEM. The merits and disadvantages of these two algorithms are highlighted. An ingeniously combined algorithm exploiting the merits of both BE and DIRK is presented and a numerical experiment shows that it can significantly improve the accuracy with no additional computing burden. For nonlinear problems, a DIRK nonlinear iteration strategy is presented and it can be shown that the total computing time of one integration time step can be shortened by about 36% without any accuracy loss in the solutions.