Flux limited schemes: Their classification and accuracy based on total variation stability regions

A classification in terms of accuracy of flux limited high resolution schemes in steep gradient region is done which is based on two different total variation (TV) stability regions. The dependence of the TV stability regions on the smoothness parameter is shown. This dependence relation relates and pave a way to define a common unified TV stability region for both class of schemes. New flux limiters, satisfying the unified TV stability region are also proposed which are robust and work efficiently for both backward (left) and forward (right) moving solution profiles. Main significant feature of this classification is that it can be used to construct hybrid schemes and improve the accuracy of all existing flux limiters based schemes. Numerical results on linear test problems are given to support the theoretical discussion.