This paper presents a new explicit scheme for the solution of shallow water equations in one and two space dimensions, developed from the space-time conservation element and solution element (CE/SE) method. The basis functions used are second-order Taylor expansions in time and space. This increase in the order of the approximation functions produces an increase in the number of unknowns in the scheme, therefore, besides the flow variables and their slopes, their second-order partial

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... are also unknown in the present scheme. An iterative process for the calculation of the first and second order derivatives is formulated for problems with shocks and discontinuities. Computational experiments demonstrate third-order accuracy. The one-dimensional and two-dimensional dam-break problems presented validate the accuracy and robustness of this scheme.