On High-Order Upwind Methods for Advection

Scheme III (piecewise linear) and V (piecewise parabolic) of Van Leer are shown to yield identical solutions provided the initial conditions are chosen in an appropriate manner. This result is counter intuitive since it is generally believed that piecewise linear and piecewise parabolic methods cannot produce the same solutions due to their different degrees of approximation. The result also shows a key connection between the approaches of discontinuous and continuous representations.