Hybrid CBSQI-WENO Schemes for Convection-Diffusion Problems

Abstract

The B-spline Quasi-Interpolation (BSQI) based numerical scheme is a successful method for obtaining the solution to partial differential equations under sufficient regularity conditions. However, it can lead to instability and spurious oscillations in the numerical solution when high gradients or discontinuities are present. To address this issue, this article proposes a hybrid version of the BSQI scheme to solve convection-diffusion problems. The hybrid scheme combines the Cubic BSQI (CBSQI) scheme with the fifth-order Weighted Essentially Non-Oscillatory (WENO) method to approximate the convective flux, and is able to compute the solution in a non-oscillatory manner. Further, we have introduced an approximate smoothness indicator for the larger stencil of the WENO scheme, derived from the smoothness indicator of the lower-order stencils. The approximate smoothness indicator is used as a troubled-cell indicator in a hybrid scheme and has allowed us to develop an efficient version of the WENO-AO(5,3) scheme (Balsara et al. J. Comp. Phy. 2016), which we call WENO-AOA(5,3) scheme. Additionally, we propose a fifth-order hybrid scheme that combines a finite-difference approximation with the WENO-AOA(5,3) scheme to solve convection-diffusion equations. To validate the proposed schemes, we conduct tests on multiple 1D and 2D cases. The hybrid schemes produce comparable results to the WENO scheme while being more computationally efficient. Specifically, the hybrid schemes are 50%–70% more efficient than the WENO-AOA(5,3) scheme, while the WENO-AOA(5,3) scheme has a 2%–15% advantage over the WENO-AO(5,3) scheme.