The flux integral method for multidimensional convection and diffusion

The jlux integral method is a procedure for constructing an explicit single-step forward-in-time conservative control-volume update of the unsteady multidimensional convection-d@usion equation. The convective-plusdifjiiveJlux at each face of a control-volume cell is estimated by integrating the transported variable and its face-normal derivative over the volume swept out by the convecting velocityJie1d. This yields a unique description of the fluxes, whereas other conservatiue methods rely on nonunique, arbitrary pseudo$uxd$erence splitting procedures. The accuracy of the resulting scheme depends on the form of the subcell interpolation assumed, given cell-average data. Cellwise constant behavior results in a (very art$cially difjiive) jirst-order convection scheme. Second-order convection-d@ision schemes correspond to cellwise linear (or bilinear) subcell interpolation. Cellwise quadratic subcell interpolants generate a highly accurate convection-diffusion scheme with excellent phase accuracy. Under constant coejicient conditions, this is a uniformly third-order polynomial interpolation algorithm (UTOPIA).