Derivation of high-order advection–diffusion schemes

Using the interpolation polynomial method, major upwind explicit advection–diffusion schemes of up to fifth-order accuracy are rederived and their properties are explored. The trend emerges that the higher the order of accuracy of an advection scheme, the easier is the task of scheme stabilization and wiggling suppression. Thus, for a certain range of the turbulent diffusion coefficient, the stability interval of third- and fifth-order up-upwind explicit schemes can be extended up to three units of the Courant number (0≤c≤3).

Having good phase behavior, advection odd-order schemes are stable within a single computational cell (0≤c≤1). By contrast, even-order schemes are stable within two consecutive grid-cells (0≤c≤2), but exhibit poor dispersive properties. Stemming from the finding that considered higher-order upwind schemes (even, in particular) can be expressed as a linear combination of two lower-order ones (odd in this case), the best qualities of odd- and even-order algorithms can be blended within mixed-order accuracy schemes. To illustrate the idea, a Second-Order Reduced Dispersion (SORD) marching scheme and Fourth-Order Reduced Dispersion (FORD) upwind scheme are developed. Computational tests demonstrate a favorable performance of the schemes. In spite of the previous practice restricting usage of even-order upwind schemes (fourth-order in particular), they exhibit a potential to stand among popular algorithms of computational hydraulics.