On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations

In this paper, a novel technique of obtaining high resolution, second order accurate, oscillation free, solution dependent weighted least-squares (SDWLS) reconstruction in finite volume method is explored. A link between the weights of the weighted least-squares based gradient estimation and various existing limiter functions used in variable reconstruction is established for one-dimensional problems for the first time. In this process, a class of solution dependent weights are derived from the link which is capable of producing oscillation free second order accurate solutions for hyperbolic systems of equations without the use of limiter function. The link also helps in unifying various independently proposed limiter functions available in the literature. The way to generate numerous new limiter functions from the link is demonstrated in the paper. An approach to verify TVD criterion of the SDWLS formulation for different choice of weights is explained. The present high resolution scheme is then extended to solve multi-dimensional problems with the interpretation of weights in SDWLS as influence coefficients. A few numerical test examples involving one- and two-dimensional problems are solved using three different new limiter functions in order to demonstrate the utility of the present approach.