A Two-Step High-Order Compact Corrected WENO Scheme

Abstract

In this study, we introduce a novel 2-step compact scheme-based high-order correction method for computational fluid dynamics (CFD). Unlike traditional single-formula-based schemes, our proposed approach refines flux function values by leveraging results from high-order compact schemes on the same stencils, provided a certain smoothness condition is met. By applying this method, we achieve a more stable and efficient compact corrected Weighted Essentially Non-Oscillatory (WENO) scheme. The results demonstrate significant improvements across all enhanced schemes, particularly in capturing shock waves sharply and maintaining stability in complex scenarios, such as two interacting blast waves, as validated through 1D benchmark tests. In addition, error analysis is also provided for the two different correction configurations based on WENO.

1. Introduction

With the advancement of computing, the demand for high-order numerical methods in numerical simulations has been steadily increasing. These methods serve as essential tools for simulating complex physical phenomena and are widely applied across various scientific and engineering fields. Over the past two decades, significant progress has been made in this domain, resulting in the development of various high-order numerical methods.

Based on finite element principles, discontinuous Galerkin (DG) methods [1,2,3] have emerged as a flexible framework for high-order discretization. These methods divide the computational domain into discrete elements and use polynomial approximations within each element to effectively represent solution gradients and discontinuities. By introducing discontinuities at element interfaces and employing suitable numerical fluxes, DG methods excel in handling shocks, making them ideal for simulating compressible flows, turbulence modeling, and multi-phase flows.

Spectral element (SE) methods [4] represent another class of high-order numerical techniques that achieve exceptional solution accuracy by utilizing the spectral precision of polynomial approximations within elements. By combining spectral accuracy with the flexibility of element-based discretization, SE methods excel in applications such as turbulence simulations, aerodynamics, and environmental fluid dynamics. Spectral volume methods and spectral difference methods also belong to the category of spectral-based high-order numerical techniques. Similarly, spectral volume methods (SVMs) [5,6] combine spectral accuracy with control volume principles to precisely represent fluxes and gradients within computational cells. Other spectral difference methods (SDMs) [7,8], on the other hand, integrate spectral approximations with finite-difference methods, achieving a balance between computational efficiency and accuracy. Both approaches are effective in simulating phenomena such as turbulent flows, shock waves, and combustion processes.

Group velocity control methods [9] offer a unique approach, focusing on regulating the propagation characteristics of numerical methods to enhance stability and accuracy. By adjusting the group velocity within the discretization framework, these methods reduce dispersion and dissipation errors, thereby improving the fidelity of wave propagation simulations. These techniques are applicable in fields such as acoustics, electromagnetics, and solid mechanics.

While the aforementioned methods excel in achieving extreme accuracy and handling discontinuity problems, simulating high-speed, complex flows requires numerical methods that not only attain high precision to capture fine structures in flow fields but are also robust enough to handle strong discontinuities, such as shock waves. Essentially, non-oscillatory (ENO) methods and their weighted variants (WENO) [10,11,12,13,14,15], based on traditional finite difference, are still the practical way for most high-speed complex flow simulations. These high-order methods provide precise techniques for the discretization of partial differential equations, enabling accurate analysis of complex physical mechanisms, the capture of intricate wave propagation, and the handling of phenomena characterized by steep gradients and discontinuities. WENO methods, based on ENO principles, stand out for their stability and ability to suppress oscillations when handling discontinuities. By employing multiple stencils and smoothness-based weighted reconstructions, WENO methods achieve a balance between accuracy and stability, proving effective in scenarios involving shock waves and other discontinuities. Additionally, Deng et al. [16,17] proposed a fifth-order shock-capturing scheme with a two-stage boundary variation diminishing (BVD) algorithm with an improved approach combining upwind-biased interpolations to further enhance the capability of handling discontinuities, offering robust performance for high-speed flow simulations. Jin et al. [18] also proposed a new family of WENO schemes which satisfy the scale-invariant property.

To tackle greater challenges in numerical simulations of complex high-speed fluid dynamics, combustion, and other intricate flow phenomena, researchers have further enhanced WENO by developing various variants. For instance, WENO-JS [14] introduced a novel smoothness indicator based on Lagrange interpolation polynomials. WENO-Z [19] refined this methodology by employing a global smoothness indicator derived from WENO-JS stencils, reducing numerical dissipation near shocks while preserving high-order accuracy. Castro et al. [20] extended the framework of smoothness indicators, enabling WENO-Z to achieve arbitrary odd-order precision. Ha et al. [21] proposed a new smoothness indicator to develop a fifth-order WENO-Z method tailored specifically for first-order critical points. Acker et al. [22] presented WENO-Z+, which improves numerical resolution by optimizing less smooth substencils. Wang et al. [23] identified limitations in the accuracy of fifth-order WENO-Z methods at high-order critical points and proposed WENO-D—a fifth-order variant that incorporates a corrective function to enhance convergence accuracy relative to the weight functions used in WENO-Z. The Central WENO (CWENO) method [24,25] prioritizes the central stencil as the most stable option under smooth flow conditions, consistently showcasing its reliability. For non-smooth flows, smoothness indicators guide the reconstruction process to select the optimal one-sided stencil. Additionally, the adaptive order WENO (WENO-AO) method [26] integrates a fourth-degree polynomial and three quadratic polynomials into a new adaptive fifth-order approach, utilizing Legendre polynomials to calculate smoothness indicators. However, most of these improvements are based on extra stencils, leading to an increase in calculation. Meanwhile, using more points to calculate the derivatives near the strong discontinuities may also lead to inaccurate results.

The Weighted Compact Scheme (WCS), proposed by Jiang, Shan, and Liu [27], integrates the principles of WENO and compact schemes [28,29,30]. By employing Hermite polynomials instead of Lagrange polynomials within each stencil, WCS achieves enhanced accuracy. In shock regions, WCS effectively adjusts stencil contributions to limit the influence of those containing discontinuities. However, the original formulation of WCS faces instability issues when addressing strong discontinuities and fails to sufficiently resolve global dependency problems associated with derivatives in compact schemes. To address these challenges, Fu et al. [31] introduced the Modified Weighted Compact Scheme (MWCS). This method combines results from WENO and WCS, assigning appropriate weights to both. The MWCS eliminates oscillations while enabling local computation of weight coefficients for pressure and density at each time step. However, compared to traditional WENO methods, the MWCS incurs additional computational costs due to it solving a five-diagonal matrix system, effectively doubling the workload. Furthermore, stability issues arising from compact schemes remain unresolved. In addition, this leads to the rapid propagation of boundary errors throughout the flow field.

In our previous study [32], we modified and improved the linear scheme used in the central stencil in WENO by incorporating a high-order compact scheme inspired by the MWCS, enhancing computational accuracy with the same points as used in the fifth-order WENO scheme. In this study, we further investigate this improvement and propose another correction method which enhances all stencils in the WENO scheme. We compare the numerical solutions obtained from these two new schemes, perform error analysis for both, compare the results with the results from WENO with characteristic flux, and propose a two-step high-order correction method applicable to more general scenarios.

The structure of this paper is as follows: Section 2 introduces the numerical methods and formulas used in the study; Section 3 presents the numerical simulation results for three benchmark cases (Sod, Shu–Osher, and two interacting blast waves) and provides a comparison and discussion of the numerical results obtained using the original WENO, the MWCS, and the WENO with characteristic flux. Section 4 includes error analysis and compares the computational time required. Finally, Section 5 summarizes the findings and provides concluding remarks.

2. Numerical Methods

This section primarily outlines concepts of high-order conservative difference schemes and the WENO scheme, as well as high-order correction methods based on the compact scheme. Without loss of generality, we begin with the scalar conservation equation in the one-dimensional case.

$$\frac{\partial }{\partial t}u\left(x,t\right)=\frac{\partial }{\partial x}f\left(u\left(x,t\right)\right)$$

(1)

The high-order numerical schemes focus on the spatial discretization on the right-hand side of Equation (1), utilizing the discrete grid points (evenly distributed) in Figure 1. The normal grid points are marked as circles with integer indices (j − 2, j − 1, j, j + 1, j + 2), while the indices for the midpoints are j − 3/2, j − 1/2, j + 1/2, and j + 3/2, respectively.

2.1. Conservative Scheme

To obtain the derivative at the j-th point (or $x_j$), the numerical flux $\widehat{f}$ defined at the midpoints are used in the following equation to ensure the conservation:

$$f_j^{\text{′}}=\frac{\partial }{\partial x}f\left(x_j\right)=\frac{{\widehat{f}}_{j+\frac{1}{2}}-{\widehat{f}}_{j-\frac{1}{2}}}{h_j}$$

(2)

The flux function can be calculated using the values of original function $f$ through an implicit definition as follows:

$${\widehat{f}}_j=\widehat{f}\left(u\left(x_j, t\right)\right) \equiv \frac{1}{h_j}{{\displaystyle \int }}_{x_{j-\frac{1}{2}}}^{x_{j+\frac{1}{2}}}f\left(\xi \right)d\xi .$$

(3)

where $x_{j-\frac{1}{2}}$ is the midpoint between $x_{j-1}$ and $x_j$ and $h_j=x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}$.

We can further define a primitive function $H$ of $f\left(\xi \right)$, and its value at the midpoints can be calculated by

$$H_{j+\frac{1}{2}}=H\left(x_{j+\frac{1}{2}}\right)={{\displaystyle \int }}_{-\infty }^{x_{j+\frac{1}{2}}}f\left(\xi \right)d\xi ={\displaystyle \sum }_{i=-\infty }^j{{\displaystyle \int }}_{x_{i-\frac{1}{2}}}^{x_{i+\frac{1}{2}}}f\left(\xi \right)d\xi ={\displaystyle \sum }_{i=-\infty }^j{\widehat{f}}_i{h}_i.$$

(4)

Then, the derivative of the primitive function at the midpoint equals the flux function value; in other words,

$${H^{\prime }}_{j+\frac{1}{2}}={\widehat{f}}_{j+\frac{1}{2}}.$$

(5)

Substituting Equation (5) into Equation (2) yields

$$f_j^{\text{′}}=\frac{H_{j+\frac{1}{2}}^{\text{′}}-H_{j-\frac{1}{2}}^{\text{′}}}{h_j}.$$

(6)

More details regarding the conservative difference schemes and the flux functions can be found in Reference [4].

2.2. The 5th-Order WENO Scheme

The core principle of the WENO scheme lies in constructing a high-order numerical flux approximation through a nonlinear weighted combination of several lower-order schemes on substencils. These weights are dynamically adjusted based on the smoothness properties of the function within each stencil. To achieve a fifth-order WENO scheme, three second-order flux approximations for ${\widehat{f}}_{j+\frac{1}{2}}$ are generated from function values at the grids on substencils, as illustrated in Figure 1

$$E_0=\left\{ f_{j-2}, f_{j-1}, f_j\right\}, E_1=\left\{ f_{j-1}, f_j, f_{j+1}\right\}, E_2=\left\{ f_j, f_{j+1}, f_{j+2}\right\}.$$

The three second-order flux approximations for ${\widehat{f}}_{j+\frac{1}{2}}$ are

$$\begin{gathered} E_0: {\widehat{f}}_{j+\frac{1}{2}}=\frac{1}{3}{\widehat{f}}_{j-2}-\frac{7}{6}{\widehat{f}}_{j-1}+\frac{11}{6}{\widehat{f}}_j \\ {E}_1: {\widehat{f}}_{j+\frac{1}{2}}=-\frac{1}{6}{\widehat{f}}_{j-1}+\frac{5}{6}{\widehat{f}}_j+\frac{1}{3}{\widehat{f}}_{j+1} \\ {E}_2: {\widehat{f}}_{j+\frac{1}{2}}=\frac{1}{3}{\widehat{f}}_j+\frac{5}{6}{\widehat{f}}_{j+1}-\frac{1}{6}{\widehat{f}}_{j+2}. \end{gathered}$$

(7)

The weighted average of the above lower-order approximations with optimal weights based on the smoothness of the stencils is

$${\widehat{f}}_{j+\frac{1}{2}}={\omega }_0{\widehat{f}}_{j+\frac{1}{2}}^{\left(E_0\right)}+{\omega }_1{\widehat{f}}_{j+\frac{1}{2}}^{\left(E_1\right)}+{\omega }_2{\widehat{f}}_{j+\frac{1}{2}}^{\left(E_2\right)},$$

(8)

where

$${\omega }_i=\frac{{\gamma }_i}{{\gamma }_0+{\gamma }_1+{\gamma }_2}\mathrm{and} {\gamma }_i=\frac{c_i}{{\left(\epsilon +I{S}_i\right)}^2},$$

(9)

and $c_0=\frac{1}{10}, c_1=\frac{3}{5},$ and $c_2=\frac{3}{10}$. $I{S}_i$ represents the smoothness indicators and can be found in Reference [8].

2.3. Compact Corrected WENO (CCWENO)

If we construct the corresponding compact format on each substencil of the WENO format, two third-order formulas and one fourth-order formula will be obtained as follows:

$$\begin{gathered} E_0: 2{H^{\prime }}_{j-\frac{1}{2}}+{H^{\prime }}_{j+\frac{1}{2}}=\frac{1}{2h_j}\left(-H_{j-\frac{3}{2}}-4H_{j-\frac{1}{2}}+5H_{j+\frac{1}{2}}\right) \\ {E}_1: {H^{\prime }}_{j-\frac{1}{2}}+4{H^{\prime }}_{j+\frac{1}{2}}+{H^{\prime }}_{j+\frac{3}{2}}=\frac{3}{h_j}\left(H_{j+\frac{3}{2}}-H_{j-\frac{1}{2}}\right) \\ {E}_2: {H^{\prime }}_{j+\frac{1}{2}}+2{H^{\prime }}_{j+\frac{3}{2}}=\frac{1}{2h_j}\left(-5H_{j+\frac{1}{2}}+4H_{j+\frac{3}{2}}+H_{j+\frac{5}{2}}\right). \end{gathered}$$

(10)

If a scheme similar to WENO is used to merge these three low-order approximations (the weights need to be recalculated), a sixth-order accurate WCS [27] can be obtained. However, WCS encounters instability in actual use and cannot be used to capture strong discontinuity problems. In addition, WCS cannot solve the derivative value at a single point alone; rather, it needs to solve a large set of linear equations to solve the values at all points at the same time, which is a problem for controlling the propagation of boundary errors and numerical stability.

In order to solve the instability problem of WCS, the modified weighted compact format (MWCS) [31] combines the results of WENO and WCS with a certain weight, but there is still an instability problem when dealing with sharp shock waves. In addition, since the pentadiagonal linear system needs to be solved additionally, this improvement also doubles the computational workload.

Inspired by the above two methods, we proposed a new method, which uses the compact scheme to correct the WENO flux to improve the accuracy of numerical simulation. The basic idea of this method is to enhance the weight of the central template $E_1$ under certain smooth flow conditions. In this study, we further investigate this improvement and propose an alternative correction method that enhances all templates in the WENO scheme. We compare the numerical solutions of these two new methods based on flux correction using compact schemes, perform error analysis on both, compare our results with the WENO results with flux-characteristic fluxes, and propose a two-step high-order correction method applicable to more general scenarios.

The compact correction method proposed in this study uses the compact schemes in Equation (10) as a correction step. By solving the flux function at $j+\frac{1}{2}$ only, the flux functions at other points are moved to the right-hand side as follows:

$$\begin{gathered} E_0: {\widehat{f}}_{j+\frac{1}{2}}=\frac{1}{2h_j}\left(-H_{j-\frac{3}{2}}-4H_{j-\frac{1}{2}}+5H_{j+\frac{1}{2}}\right)-2{\widehat{f}}_{j-\frac{1}{2}} \\ {E}_1: {\widehat{f}}_{j+\frac{1}{2}}=\frac{3}{4h_j}\left(H_{j+\frac{3}{2}}-H_{j-\frac{1}{2}}\right)-\frac{1}{4}\left({\widehat{f}}_{j-\frac{1}{2}}+{\widehat{f}}_{j+\frac{3}{2}}\right) \\ {E}_2: {\widehat{f}}_{j+\frac{1}{2}}=\frac{1}{2h_j}\left(-5H_{j+\frac{1}{2}}+4H_{j+\frac{3}{2}}+H_{j+\frac{5}{2}}\right)-2{\widehat{f}}_{j+\frac{3}{2}}. \end{gathered}$$

(11)

The primitive functions on the right-hand side of Equation (11) can be calculated by function values at the grid points, and the flux functions can be obtained by WENO. Thus, the compact correction algorithm can be succinctly summarized as follows:

$$\begin{gathered} E_0: {\widehat{f}}_{j+\frac{1}{2}}^{cc}=\frac{1}{2h_j}\left(f_{j-1}+5f_j\right)-2{\widehat{f}}_{j-\frac{1}{2}}^{WENO} \\ E\_1: {\widehat{f}}_{j+\frac{1}{2}}^{cc}=\frac{3}{4h_j}\left(3f_j+f_{j+1}\right)-\frac{1}{4}\left({\widehat{f}}_{j-\frac{1}{2}}^{WENO}+{\widehat{f}}_{j+\frac{3}{2}}^{WENO}\right) \\ {E}_2: {\widehat{f}}_{j+\frac{1}{2}}^{cc}=\frac{1}{2h_j}\left(5f_{j+1}+f_{j+2}\right)-2{\widehat{f}}_{j+\frac{3}{2}}^{WENO}, \end{gathered}$$

(12)

where ${\widehat{f}}^{WENO}$ denotes the flux computed using the original WENO scheme, and ${\widehat{f}}^{cc}$ represents the high-order corrected flux. This method applies a high-order correction without requiring a complete recalculation of the flux function at each point, thereby avoiding exponential increases in computational cost. Since the WENO-calculated values remain incorporated in the computation, this approach prevents excessive discontinuities in the flux function. To further mitigate abrupt changes in flux values, particularly in regions of strong discontinuity, a smoothness criterion ($\beta$) is introduced, ensuring that corrections are applied only where the numerical solution at grid point ($x_j$) is sufficiently smooth. In this study, we adopt the nonlinear weight of the middle stencil from WENO as the smoothness criterion, setting $\beta ={\omega }_1$. The correction applied to the flux function at $j+\frac{1}{2}$ is given by the following equation:

$${\widehat{f}}_{j+\frac{1}{2}}=\{\begin{array}{c}{\widehat{f}}_{j+\frac{1}{2}}^{cc} if \beta \ge \alpha \\ {\widehat{f}}_{j+\frac{1}{2}}^{WENO} if \beta <\alpha \end{array}.$$

(13)

In all numerical simulations conducted in this study, $\alpha$ is set to 0.1, meaning that smoothness is weighted at 10% across all three stencils. We designate the method that applies corrections exclusively to the middle stencil as CCWENO (Compact Corrected WENO), while the approach that implements high-order compact corrections across all three stencils is referred to as CCWENO2. The parameter $\alpha$ introduced here offers an optional choice, allowing for flexible tuning of accuracy and robustness to suit specific requirements.

2.4. Introduction to the 1D Benchmark Cases

Without loss of generality, this study first employs a series of widely recognized benchmark tests based on the one-dimensional Euler equations to validate and evaluate the proposed method. These tests aim to assess the performance of numerical schemes in resolving fine-scale structures (such as turbulence) and accurately capturing significant discontinuities (such as shock waves). The one-dimensional Euler equation, regarded as a simplified version of the more complex three-dimensional Navier–Stokes equations, is expressed in vector and conservation form as follows:

$$\frac{\partial }{\partial t}\left[\begin{array}{c}\rho \\ \rho u\\ E\end{array}\right]+\frac{\partial }{\partial x}\left[\begin{array}{c}\rho u\\ \rho u^2+p\\ u\left(E+p\right)\end{array}\right]=0,$$

(14)

where $\rho$, $u$, and $E$ represent density, velocity, and total energy, respectively. $p$ represents pressure and can be calculated through $p=\left(\gamma -1\right)\left(E-\frac{1}{2}\rho u^2\right)$.

2.4.1. Sod Shock Tube Problem

To evaluate the ability to capture shocks, the Sod shock tube problem [31] is employed using the specified initial conditions

$$\left[\begin{array}{c}{\rho }_L\\ p_L\\ u_L\end{array}\right]=\left[\begin{array}{c}1.0\\ 1.0\\ 0.0\end{array}\right], \left[\begin{array}{c}{\rho }_R\\ p_R\\ u_R\end{array}\right]=\left[\begin{array}{c}0.125\\ 0.1\\ 0.0\end{array}\right].$$

(15)

Here, the subscripts L and R represent the states on the left and right sides of the domain, respectively, with the central interface positioned at ($x=0$).

2.4.2. Shu–Osher Problem

To analyze the proposed method’s ability to accurately capture shocks and their interactions with small-scale waves, the Shu–Osher problem [10] is examined. This test is known for its complexity, as numerical dissipation can significantly dampen entropy waves, making it highly sensitive. Using the same governing Equation (14), the initial conditions are defined as follows:

$$\left[\begin{array}{c}{\rho }_L\\ p_L\\ u_L\end{array}\right]=\left[\begin{array}{c}3.857143\\ 10.33333\\ 2.629369\end{array}\right], \left[\begin{array}{c}{\rho }_R\\ p_R\\ u_R\end{array}\right]=\left[\begin{array}{c}1+0.2\sin \left(5x\right)\\ 1.0\\ 0.0\end{array}\right]$$

(16)

The subscripts L and R represent the states on the left and right sides of the domain, respectively, with the central interface positioned at ($x=0$).

2.4.3. Problem of Two Interacting Blast Waves

The simulation of two interacting blast waves [33] is conducted to evaluate the ability of the method to handle strong shock waves and their multiple interactions. The initial conditions are defined as follows, and the governing equation, Equation (14), is utilized in the process:

$$\left[\begin{array}{c}{\rho }_L\\ p_L\\ u_L\end{array}\right]=\left[\begin{array}{c}1.0\\ 1000.0\\ 0.0\end{array}\right], \left[\begin{array}{c}{\rho }_M\\ p_M\\ u_M\end{array}\right]=\left[\begin{array}{c}1.0\\ 0.01\\ 0.0\end{array}\right], \left[\begin{array}{c}{\rho }_R\\ p_R\\ u_R\end{array}\right]=\left[\begin{array}{c}1.0\\ 100.0\\ 0.0\end{array}\right]$$

(17)

Here, the subscripts L, M, and R represent the conditions in the left region ($0\le x\le 0.1$), middle region ($0.1<x\le 0.9$), and right region ($0.9<x\le 1.0$), respectively.

3. Numerical Results

In the following section, we analyze the three benchmark cases mentioned above using several numerical schemes, including the fifth-order WENO scheme, the MWCS, and the two proposed schemes, CCWENO and CCWENO2, introduced in this paper. All of these schemes utilize the same five-point stencil and three substencils, ensuring a consistent basis for comparison.

Additionally, we include the numerical results of the WENO scheme incorporating characteristic-wise reconstruction, as discussed in [34]. By reconstructing the variables along characteristic directions, this approach significantly reduces spurious oscillations and numerical dissipation, which are commonly encountered in high-order schemes. However, this enhanced accuracy comes at the cost of a considerable increase in computational effort due to the added complexity of characteristic-wise operations.

In contrast, the proposed CCWENO and CCWENO2 schemes achieve comparable accuracy without the need for characteristic-wise reconstruction. This innovative feature simplifies their implementation and reduces computational overhead while maintaining robust performance in capturing complex flow features.

Figure 2 shows the numerical results for the Sod shock tube problem, a classic test case in computational fluid dynamics used to evaluate the ability of numerical schemes to resolve discontinuities and shock waves. Specifically, Figure 2b provides a magnified view of the shock discontinuity region, allowing for a closer inspection of the schemes’ accuracy and resolution at the discontinuity.

For all numerical schemes analyzed, a uniform grid size of 200 points was employed, alongside a consistent time step of $dt={10}^{-4}$, ensuring equitable comparisons across methods. To provide a benchmark reference for comparison, the WENO scheme was utilized with a significantly finer grid resolution of 1600 points, coupled with a reduced time step of $dt={10}^{-5}$, delivering enhanced spatial and temporal accuracy in the refined solution.

The results indicate that the proposed CCWENO and CCWENO2 schemes exhibit highly comparable numerical behavior, particularly in their ability to accurately resolve the shock wave discontinuity. Both schemes demonstrate a notable advantage over other methods, achieving closer alignment to the true shock wave position while maintaining high-resolution detail. These findings highlight the strengths of compact corrected WENO schemes in applications involving sharp gradients and shock waves.

Figure 3 presents the numerical results for the Shu–Osher problem, a benchmark test case designed to assess the accuracy and resolution of numerical schemes in capturing fine-scale structures and shock interactions. Figure 3b provides a magnified view of the region exhibiting significant fluctuations and the primary shock wave discontinuity, allowing for a closer examination of the schemes’ performance in resolving intricate flow features.

For all numerical schemes under consideration, we used the same grid size (200 grid points) and time accuracy (${10}^{-4}$), ensuring consistency across the simulations. The standard WENO scheme with a much higher grid density of 1600 points and smaller a time step of $dt={10}^{-5}$ was used to obtain the refined solution. The results reveal that both the CCWENO and CCWENO2 schemes excel in capturing the fine vortex structures generated in this problem. Among the two, CCWENO2 achieves a closer match to the amplitude of the refined solution, demonstrating its capability to approximate the true solution with remarkable fidelity. However, it introduces a more pronounced phase error (dispersion) compared to the CCWENO scheme, which maintains greater alignment with the reference solution in terms of phase accuracy.

Figure 4 illustrates the numerical results obtained for the challenging two interacting blast waves problem, which involves both complex wave systems and strong discontinuities. In all the tested numerical schemes, the grid size remains at 200 points, and the time step size is also kept as $dt={10}^{-4}$, as in the previous benchmark cases. The refined numerical solution for comparison is still obtained through WENO scheme with 1600 points, and $dt$ is to be reduced to ${10}^{-5}$.

The results highlight the superior performance of the proposed CCWENO and CCWENO2 schemes in capturing the intricate dynamics of the interacting blast waves. These schemes demonstrate a remarkable ability to resolve complex phenomena, including the delicate interplay between discontinuities and fluctuations. Compared to other formats, CCWENO and CCWENO2 produce sharper and more accurate representations of these features, showcasing their effectiveness in preserving essential flow characteristics.

4. Error Analysis and Efficiency Comparison

4.1. Error Analysis

In addition to the above-mentioned tests based on benchmark cases, this study also conducts error analysis based on discrete Fourier transforms (Fourier stability analysis or von Neumann stability analysis) [35]. It examines the behavior of numerical errors by breaking them down into Fourier modes and evaluating their growth or attenuation over time. This approach is instrumental in assessing whether a given CFD scheme will yield a stable and accurate solution or result in numerical instability and oscillatory behavior.

Figure 5 illustrates the dissipation (left) and dispersion (phase error, right) characteristics of the WENO scheme compared to the CCWENO and CCWENO2 schemes (positive flux) within a sufficiently smooth region ($c_0=\frac{1}{10}, c_1=\frac{3}{5},$ and $c_2=\frac{3}{10}$ are used for the weights). The results clearly demonstrate that the dissipation produced by the CCWENO and CCWENO2 schemes is significantly lower than that of the WENO scheme, highlighting their improved ability to preserve energy and accuracy in smooth flows. Furthermore, the dispersion (phase error) is also notably reduced, as evidenced by the closer alignment of the curves to the ideal reference line ($y=x$), indicating better phase accuracy and reduced numerical distortion.

Figure 6 depicts the error distribution for the three schemes within the shock wave region (the nonlinear weights are ${\omega }_0=0.45, {\omega }_1=0.45, \mathrm{and} {\omega }_2=0.1$, respectively). The results reveal that, except for the extremely small high wave number region, the dissipation generated by the CCWENO and CCWENO2 schemes is significantly lower compared to the WENO scheme. Additionally, the dispersion (phase error) is noticeably reduced, demonstrating the superior capability of the CCWENO schemes in accurately resolving shock wave dynamics while minimizing numerical distortions.

Figure 7 illustrates the outcome obtained after further reducing the smoothing weight applied to the third stencil (${\omega }_0=0.495, {\omega }_1=0.495, \mathrm{and} {\omega }_2=0.01$,respectively). The results indicate that this adjustment yields a solution that is nearly identical to the one presented in Figure 6, demonstrating minimal variation despite the modification.

4.2. Efficiency Comparison

Figure 8 presents a detailed comparison of the running times (in seconds) for five schemes in solving the two interacting blast waves problem: the original WENO scheme, WENO with characteristic-wise reconstruction, the MWCS, and the proposed CCWENO and CCWENO2 schemes. The computations were executed on the same personal computer equipped with an AMD Ryzen 7 5700U processor, ensuring reliable performance metrics.

The analysis reveals that the CCWENO scheme incurs only a modest increase in computational effort compared to the standard WENO scheme without characteristic-wise reconstruction. However, it demonstrates significantly reduced running time when benchmarked against the MWCS and WENO with characteristic-wise reconstruction schemes, making it a more efficient choice overall. On the other hand, while the numerical results of CCWENO2 closely resemble those obtained by CCWENO in terms of accuracy and solution quality, CCWENO2 demands comparatively higher computational time.

These findings underscore the computational efficiency of the CCWENO scheme, particularly in scenarios where a balance between accuracy and runtime performance is critical. Meanwhile, the CCWENO2 scheme offers similar numerical reliability but introduces a trade-off in terms of runtime, which may influence its applicability depending on specific use-case requirements.

5. Conclusions

This study introduces an innovative high-order correction methodology for CFD, built upon a two-step compact scheme framework. Unlike conventional single-formula approaches, the proposed method enhances flux function calculations by incorporating high-order compact schemes over identical stencils. This strategy ensures improved accuracy in regions with significant flow variations. The application of this new method leads to the development of a more stable and efficient compact corrected WENO schemes. The proposed CCWENO scheme demonstrates substantial performance improvements, particularly in sharply resolving shock waves and maintaining numerical stability in complex flow scenarios, and it also shows the excellent efficiency compared to other methods. These advancements have been rigorously validated using one-dimensional benchmark tests. Moreover, this study included a thorough error analysis for the two proposed schemes. This analysis further solidifies the practical utility of the proposed method in high-fidelity CFD simulations. Finally, in contrast to purely compact-based schemes, this method or scheme has the flexibility to be applied selectively rather than across the entire computational domain. By focusing on specific regions where higher accuracy is essential, it can further optimize computational efficiency while ensuring precision in critical areas.