A hybrid first-order and WENO scheme for the high-resolution and computationally efficient modeling of pollutant transport

Highlights

• A hybrid first-order and WENO scheme is proposed for pollutant transport modeling.

• The new model achieves comparable quantitative accuracy as compared to the WENO method but has significantly reduced computational cost.

• A threshold value for switching between first-order and WENO schemes is recommended.

Abstract

To improve the modeling quality of pollutant transport in shallow waters, different reconstruction schemes have been proposed to better link the edge values to the centroid values of a pollutant concentration in finite-volume shallow water models: a scheme of higher (lower) order generally has a better (poorer) quantitative accuracy but lower (higher) computational efficiency. Here, a numerical comparative study of several classical schemes is first conducted under a variety of pollutant distribution conditions. The results reveal that, for the condition of relatively uniform pollutant distribution, the numerical accuracy of a lower-order scheme (such as the first-order scheme or the MUSCL scheme) may be similar to that of a higher-order scheme (such as the WENO scheme). The second-order derivative of the concentration, here termed the nonlinear indicator (NI), correlates well with the discrepancies between the numerical solutions and analytical solutions. A threshold value of approximately ${10}^{-7}\sim {10}^{-6}$ m^-2 for the NI is identified, above which a higher-order scheme may be required. Based on this understanding, a hybrid first-order and WENO scheme is proposed. Numerical case studies show that the hybrid scheme can successfully combine the efficiency of the first-order scheme with the high accuracy of the WENO scheme for pollutant modeling.

Introduction

Reliable and efficient numerical modeling of pollution transport plays an important role in environmental protection. In this regard, the depth-averaged advection-diffusion-reaction equation for pollutants and the shallow water equations have been widely applied. Many different numerical methods have been used to solve these equations [2,6,8,21,24]. In particular, the Godunov-type finite volume method has received increasing attention, possibly owing to 1) the increasing importance of pollutant transport in extreme flow conditions and 2) the capability of the Godunov-type finite volume method to automatically capture shock waves/contact discontinuities [38]. Physically and mathematically, this advantage is achieved by treating the numerical flux estimation as a Riemann problem using exact or approximate Riemann solvers. Once the specific Riemann solver has been chosen, the key is to reconstruct the edge values from the centroid values of the pollutant concentration, which serve as independent variables in Riemann solvers. Therefore, an accurate reconstruction scheme that links the edge values to the centroid values is vital for high-quality pollutant modeling.

The most straightforward reconstruction method is the first-order scheme, which directly sets the edge values equal to the centroid values. Since the first-order scheme assumes a uniform concentration distribution within computational cells, it is rarely used [6,14,19]. Various 2nd-order MUSCL (Monotone upstream-centered schemes for conservation laws) schemes were introduced by assuming a linear variation in the concentration between the cell center and the cell edge [4,[9], [10], [11], 42]. Nevertheless, conditions with very nonlinear variations may still induce considerable numerical diffusion. For example, Benkhaldoun et al. [5] reported a loss of approximately 78% in the peak concentration values when simulating pollutant transport in a squared cavity. Using a higher-order reconstruction scheme, such as the high-order accurate WENO (weighted essentially nonoscillatory) scheme, may be a feasible way to overcome this drawback [30], [43]. However, a WENO scheme usually requires much more CPU time [36] and becomes advantageous only when facing complex pollutant distributions. This paper proposes a hybrid scheme to make use of both the high efficiency of a low-order scheme in regions with relatively smooth pollutant distributions and the high accuracy of a high-order scheme in regions with complex pollutant distributions. The concept of a hybrid scheme can also be seen in the modeling of the compressible turbulence problems [7,20,23]. For hybrid schemes, the most important parameter is a quantitative switch indicator that can effectively discriminate regions with relatively smooth and complex pollutant distributions [31], [32], [33], [34]. In this regard, a numerical comparative study of several classical schemes is conducted under a variety of pollutant spatial distribution conditions, from which it is shown that the second-order derivative of the pollutant concentration, termed here the nonlinear indicator (NI), serves as an appropriate switch indicator. A threshold value for the NI is then recommended based on the variation trend of the discrepancy of the numerical solutions and analytical solutions against the NI. Using this switch indicator, the hybrid first-order and WENO scheme is proposed, of which the performance is demonstrated by efficiency comparisons and a convergence rate study. This paper is organized as follows. In Section 2, we present the modeling framework of the finite-volume shallow water and pollutant transport model, within which several classical reconstruction schemes as well as the conceptualized hybrid scheme are introduced. In Section 3, we analyze the correlation between the numerical error and nonlinear characteristic and propose the hybrid first-order-WENO scheme. In Section 4, a systematic investigation of the performance of the hybrid scheme is conducted. Section 5 summarizes the paper with some concluding remarks.