Evaluation of various high‐order‐accuracy schemes with and without flux limiters

Abstract

AbstractConventional high‐order schemes with reduced levels of numerical diffusion produce results with spurious oscillations in areas where steep velocity gradients exist. To prevent the development of non‐physical oscillations in the solution, several monotonic schemes have been proposed. In this work, three monotonic schemes, namely Van Leer's scheme, Roe's flux limiter and the third‐order SHARP scheme, are compared and evaluated against schemes without flux limiters. The latter schemes include the standard first‐order upwind scheme, the second‐order upwind scheme and the QUICK scheme. All the above schemes are applied to four two‐dimensional problems: (i) rotation of a scalar ‘cone’ field, (ii) transport of a scalar ‘square’ field, (iii) mixing of a cold with a hot front and (iv) deformation of a scalar ‘cone’ field. These problems test the ability of the selected schemes to produce oscillation‐free and accurate results in critical convective situations. The evaluation of the schemes is based on several aspects, such as accuracy, economy and complexity. The tests performed in this work reveal the merits and demerits of each scheme. It is concluded that high‐order schemes with flux limiters can significantly improve the accuracy of the results.