High resolution schemes

Abstract

This chapter continues the development of convection schemes and discusses approaches by which boundedness is enforced on High Order (HO) convection schemes to produce High Resolution (HR) schemes. The recipe for a HR scheme is shown to involve a combination of a HO profile and a Convection Boundedness Criterion (CBC) ensuring that no oscillatory behavior is experienced in the solution. The Normalized Variable Formulation (NVF) and the Total Variation Diminishing (TVD) frameworks for developing HR schemes are introduced. Even though the two approaches look very different they are shown to be almost identical. The Normalized Variable Diagram (NVD) and Sweby’s (or r - ψ) diagram for visualizing HR schemes in the NVF and TVD formulation, respectively, are presented. The functional relationships for several HR schemes are specified in the context of both the NVF and TVD formulations. In addition to the Deferred Correction (DC) procedure discussed in the previous chapter, two additional techniques for the implementation of HO and HR schemes in structured and unstructured grids are introduced, namely the Downwind Weighing Factor (DWF) method and the Normalized Weighing Factor (NWF) method.