Abstract
This note analyzes a simple discontinuous solution to hyperbolic 2 × 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point where the characteristic speeds are equal. We study the Hugoniot curves in Schaeffer & Shearer's case I and II which are relevant to the three-phase Buckley–Leverett model for oil reservoir flow. The compressive and overcompressive parts are determined. The wave curves through the umbilic point are discussed and their compressive and overcompressive parts are also determined.
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