HIGHER-ORDER TRANSPORT EQUATIONS FOR CONTROL VOLUMES OF FINITE SIZES

We present an extension to the conventional transport ¹ equation by relaxing the first-order variation assumption within a control volume of finite size². It must be clear that it is not the intention of this article to present any new higher-order scheme. By allowing higher-order variations within a control volume, we can derive second-order, and higher-order transport equations. This is particularly the case when a control volume is not infinitesimally small such that the variation within the control volume cannot be approximated as linear. This is the case in many numerical modeling exercises where the control volumes are large. Using the finite-volume method as a tool, we show that the central difference scheme and the second-order upwind scheme are natural outcomes of a second-order transport equation.We also show that the central differenceschemeindeed converts the first-order transport equation into a second-order transport equation. It is our wish and hope that this approach may lead to bounded higher-order difference scheme(s), and also allow incorporation of true physical phenomena which a first-order transport equation cannot accommodate.