The nirvana scheme applied to one‐dimensional advection

Abstract

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‐Δt generalisation of the universal limited. More restrictive TVD constraints are also extended to large Δt. 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.

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“…These effects are still evident, however, by the fact that, in the presence of a southward current, some non-zero tracer values appear to propagate northwards. Additional improvements could be obtained by increasing the order of the advection scheme and including a constraining limiter in the scheme (Leonard et al, 1995); however, this would require major alterations to the model code to improve the numerics and is not the focus of this research.…”

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