Unified Formulation for High-Order Streamline Tracing on Two-Dimensional Unstructured Grids

Accurate streamline tracing and travel time computation are essential ingredients of streamline methods for groundwater transport and petroleum reservoir simulation. In this paper we present a unified formulation for the development of high-order accurate streamline tracing algorithms on unstructured triangular and quadrilateral grids. The main result of this paper is the identification of velocity spaces that are suitable for streamline tracing. The essential requirement is that the divergence-free part of the velocity must induce a stream function. We recognize several classes of velocity spaces satisfying this requirement from the theory of mixed finite element methods and, for each class, we obtain the precise functional form of the stream function. Not surprisingly, the most widely used tracing algorithm (Pollock's method) emanates in fact from the lowest-order admissible velocity approximation. Therefore, we provide a sound theoretical justification for the low-order algorithms currently in use, and we show how to achieve higher-order accuracy both in the streamline tracing and the travel time computation.