NORMALIZED VARIABLE AND SPACE FORMULATION METHODOLOGY FOR HIGH-RESOLUTION SCHEMES

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Abstract
The normalized variable formulation (NVF) methodology of Leonard [1] provides the proper framework for the development and analysis of high-resolution convection-diffusion schemes, which combine the accuracy of higher-order schemes with the stability and boundedness of the first-order upwind scheme. However, in its current form the NVF methodology helps in deriving connective schemes for uniformly or nearly uniformly discretized spaces. To remove this shortcoming, a new, normalized variable and space formulation (NVSF) methodology is developed. In the newly developed technique, spatial parameters are introduced so as to extend the applicability of the NVF methodology to nonuniformly discretized domains. Furthermore, the required conditions for accuracy and boundedness of connective schemes on nonuniform grids are also derived. Several schemes formulated using NVF are generalized to nonuniform grids using the suggested method. Both formulations are tested on nonuniform grids by solving two problems. Computational results show substantial improvement in accuracy when using the NVSF methodology with third-order high-resolution schemes.

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