This paper introduces a new method, which we call the Mobius scheme, for the numerical solution of matrix Riccati differential equations. The approach is based on viewing the Riccati equation in its natural geometric setting, as a flow on the Grassmanian of m-planes in an (n+m)-dimensional vector space. Since the Grassmanians are compact differentiable manifolds, and the coefficients of the equation are assumed continuous, there are no singularities or intrinsic instabilities in the associated flow. The presence of singularities and numerical instabilitites is an artefact of the coordinate system, but since the Mobius scheme is based on the natural geometry, it is able to deal with numerical instability and pass accurately through the singularities. A number of examples are given to demonstrate these properties.

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