THE NIRVANA SCHEME APPLIED TO ONE-DIMENSIONAL ADVECTION

The NIRVANA project is concerned with the development of a nonoscillatory, integrally reconstructed, volume-averaged numerical advection scheme. The conservative, flux-based finite-volume algorithm is built on an explicit, single-step, forward-in-time update of the cell-average variable, without restrictions on the size of the time-step. There are similarities with semi-Lagrangian schemes; a major difference is the introduction of a discrete integral variable, guaranteeing conservation. The crucial step is the interpolation of this variable, which is used in the calculation of the fluxes; the (analytic) derivative of the interpolant then gives sub-cell behaviour of the advected variable. In this paper, basic principles are described, using the simplest possible conditions: pure one-dimensional advection at constant velocity on a uniform grid, Piecewise Nth-degree polynomial interpolation of the discrete integral variable leads to an Nth-order advection scheme, in both space and time. Nonoscillatory results correspond to convexity preservation in the integrated variable, leading naturally to a large-At generalisation of the universal limited. More restrictive TVD constraints are also extended to large Ar, Automatic compressive enhancement of step-like profiles can be achieved without exciting 'stair-casing'. One-dimensional simulations are shown for a number of different interpolations. In particular, convexity-limited cubic-spline and higher-order polynomial schemes give very sharp, nonoscillatory results at any Courant number, without clipping of extrema. Some practical generalisations are briefly discussed.