Hybrid LUP-WENO scheme for hyperbolic systems of conservation laws with troubled cell detection

Source abstract

Abstract This work presents a high-resolution third-order hybrid finite difference weighted essentially non-oscillatory (WENO) scheme to solve hyperbolic conservation laws. The proposed hybrid algorithm is two-fold. First, we divide the computational domain into smooth and non-smooth parts based on a troubled cell indicator. We design troubled cell indicators based on smoothness indicators of the WENO scheme introduced in Arun et al (2024 Int. J. Numer. Methods Fluids. 96 44–86). Next, we introduce a novel and simple switching mechanism between WENO and a linear upwind scheme to detect smooth extrema. The switching mechanism searches for the smooth extrema in such a way that it implements the linear high-order reconstructed numerical flux if the smooth extrema lie outside the stencil and a WENO procedure to compute numerical flux if the smooth extrema lie inside the stencil. We implement the TVD-RK3 scheme for time integration. The advantage of this new scheme lies in its robustness, efficiency, and high resolution. Several numerical test cases from the scalar equations to systems of Euler equations are presented to demonstrate the performance of the proposed hybrid method. The proposed scheme achieves the optimal order of convergence even in the presence of critical points while being computationally efficient. In one-dimensional problems, it reduces the computational cost by approximately 30% to 70% and by 40% to 50% in two-dimensional problems compared to existing schemes reported in the literature.