# Parallel Accelerated Fifth-Order WENO Scheme-Based Pipeline Transient Flow Solution Model
## 1 Introduction  
The water hammer pressure wave generated when a pipeline suddenly closes a valve can cause pipeline vibration, damage the valve, or cause a pipeline burst in severe cases, resulting in significant engineering accidents. Simulating, predicting, and controlling the water hammer phenomenon are critical research topics in relevant fields [1, 2, 3]. The commonly used solution schemes, including the MOC and the finite difference method, attempt to calculate water hammers [4, 5]. The MOC requires a regular grid, while the spatial step and time step are strictly constrained by the characteristic line, limiting its applicability. Many scholars have improved the classical difference solution schemes. [PERSON], [PERSON] and [PERSON] employed the MacCormack scheme to construct an explicit finite difference scheme for solving the pipeline water hammer problem [6]. They found that the second-order finite difference scheme can achieve a computational accuracy higher than the MOC. [PERSON] and others applied the MacCormack scheme of variable grids to transient flow simulation [7]. The mentioned scheme adopts the two-step prediction correction method to realize the flexible division of the spatial grid. [PERSON] and [PERSON] proposed an implicit finite-difference scheme for solving the water hammer problem in pipe network systems [8]. Although the main advantage of this scheme is that it remains stable for significant time steps, it requires a time-consuming iteration process. [PERSON] proposed a scheme for solving hyperbolic partial differential equations and proved that this scheme has high resolution and no spurious oscillations at the discontinuities [4]. The finite volume method has been widely utilized in recent years for solving water hammer waves in pipelines [9]. It can provide better conservation performance and less computational time than the TVD (Total Variation Diminishing) difference scheme. [PERSON] and [PERSON] constructed the first-order and second-order finite volume methods based on the Godunov scheme using the MUSCL-Hancock method to solve the water hammer problem [10]. They indicated that the second-order accuracy has better conservation at a low Courant number. [PERSON] proposed the prediction-correction MUSCL-Hancock finite volume method, which guarantees the accuracy of the scheme while ensuring good stability [11]. However, the second-order scheme is still limited in capturing accuracy for this aspect of strongly excited intermittent waves, and better computational results can be obtained using a higher-order scheme. The high-order ENO and WENO schemes significantly improve the capability for intermittent wave capturing [12]. The ENO scheme selects only one template among the multiple templates as the computational template, leading to inefficient computation [13]. In contrast, the WENO scheme utilizes all the templates by assigning corresponding weights to different templates through the convex combination between different templates to improve the intermittent capture performance without causing the waste of templates [14; 15; 16; 17]. For example, [PERSON] et al. [18] obtained a new global smoothness index through the Taylor expansion of the local smoothness index. In the framework of the traditional WENO-Z format, an improved third-order finite-difference WENO format was proposed. [PERSON] et al. [19] established an improved third-order finite-difference WENO scheme based on convex combinations of different polynomials to calculate the numerical fluxes at the cell boundaries and its truncation errors and parametrization are smaller than those of other third-order WENO schemes. Moreover, the numerical results indicated that the proposed scheme performs better than other third-order WENO schemes. To judge the performance of the model, in addition to the calculation accuracy of the model, the calculation efficiency of the model is also crucial.