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For many enhanced recovery methods propagating fronts arise that may be steep or discontinuous. One example is the waterflooding of a petroleum reservoir, in which there is forced out residual oil that remains after outflow by decompression has declined. In this paper high-resolution numerical methods to solve porous flow problems having propagating discontinuities are discussed. The random choice method can track solution discontinuities sharply and accurately for one space dimension. The first phase of this study adapted this method for solving the Buckley-Leverett equation for immiscible displacement in one space dimension. Extensions to more than one space dimension for the random choice method were carried out subsequently by means of fractional splitting. Because inaccuracies could be introduced for some problems at dicontinuity fronts propagating obliquely to the splitting directions, efforts are currently being directed at investigating alternatives for multidimensional cases. (ERA citation 08:004116)