Design principles for bounded higher-order convection schemes – a unified approach

The design of bounded, higher-order convection schemes is considered with a view to selecting those discretizations giving good resolution of sharp gradients, while at the same time providing competitive accuracy and convergence behaviour when applied to smooth, recirculating flows. The present work contains a detailed classification and analysis and extensive tables of most non-linear scalar convection schemes so far proposed within the cell-centred, finite-volume framework. The analysis includes a review and comparison of the two most frequently-used non-linear approaches, flux limiters (FL) and normalized variables (NV), and the three major boundedness criteria typically employed: total-variation diminishing (TVD), positivity and the convection-boundedness criterion (CBC). All NV schemes considered are converted to FL form to allow direct comparison and classification of a wide range of schemes. Several specific design principles for positive non-linear schemes are considered and it is shown how these can be applied to understand the relative performance of different approaches. Finally the performance of many existing schemes is compared and ranked on the basis of two scalar convection test cases, one smooth and one discontinuous, which demonstrates the wide variation in both accuracy and convergence behaviour of the various schemes and the benefits of the design principles considered.