In this paper, time step integration algorithms for linear first order equations with both the initial and final conditions weakly enforced are investigated. Discontinuous jumps may appear at the beginning and at the end of a time interval under consideration. The initial conditions are usually given while the final conditions are artificial variables as in the hybrid finite element formulation. If the approximate solution within a time interval is assumed to be a polynomial of degree n, there are n + 2 unknowns in the formulation. It is shown that the order of accuracy of the approximate solution would be at least n in general. If the weighting parameters (and hence the weighting functions) are chosen carefully, the order of accuracy of the approximate solution at the end of a time interval given by the final condition can be improved to 2n + 2. Besides, unconditionally stable algorithms equivalent to the generalized Pade approximations can be constructed systematically. The time-discontinuous Galerkin and bi-discontinuous Galerkin methods are treated as special cases. The weighting parameters and the corresponding weighting functions are given explicitly. Furthermore, it is shown that the accuracy of the particular solutions is compatible with the homogenous solutions if the proposed weighting functions are employed.
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