Development of High-Resolution Total Variation Diminishing Scheme for Linear Hyperbolic Problems

Abstract

A high-resolution, total variation diminishing (TVD) stable scheme is derived for scalar hyperbolic problems using the method of flux limiters. The scheme was constructed by combining the 1st-order upwind scheme and the 3rd-order quadratic upstream interpolation scheme (QUICK) using new flux limiter function. The new flux limiter function was established by imposing several conditions to ensure the TVD properties of the scheme. For temporal discretization, the theta method was used, and values for the parameter θ were chosen such that the scheme is unconditionally stable. Numerical results are presented for one-dimensional pure advection problems with smooth and discontinuous initial conditions and are compared to those of other known numerical schemes. The results show that the proposed numerical method is stable and of higher order than other common schemes.

A solver for the two-phase two-fluid model based on high-resolution total variation diminishing scheme

“…There are many numerical schemes one can apply to calculate these donored quantities. For this paper, results from 1st-order upwind scheme and flux limiter schemes (including the new scheme developed in (Abu Saleem and Kozlowski, 2015)) are shown. This will be discussed in Section 5.…”

“…There are many different forms of flux limiters. In this paper, the new scheme developed in (Abu Saleem and Kozlowski, 2015) is investigated. For this scheme, the flux limiter is given by:…”

“…The development of this scheme was based on a combination of the 1st-order upwind scheme and the 3rd-order QUICK scheme (Abu Saleem and Kozlowski, 2015).…”

“…Numerical results from this simulation are shown for the 1st-order upwind scheme and flux limiter schemes, including the new scheme derived in (Abu Saleem and Kozlowski, 2015).…”

“…We present the discrete equations of the model along with its closure relations. Temporal discretization is based on the theta method with different schemes implemented for spatial discretization including the new scheme developed in (Abu Saleem and Kozlowski, 2015). The solver is then tested and verified using numerical benchmarks and analytical solutions.…”