Novel Hybrid Unequal-Sized WENO Scheme Employing Trigonometric Polynomials for Solving Hyperbolic Conservation Laws on Structured Grids

Abstract

This study presents a novel fifth-order unequal-sized trigonometric weighted essentially non-oscillatory (US-TWENO) scheme and a novel hybrid US-TWENO (HUS-TWENO) scheme with a novel troubled cell indicator in a finite difference framework to address hyperbolic conservation laws on structured grids. Firstly, we propose three unequal-degree reconstruction polynomials in the new trigonometric polynomial space to devise a novel fifth-order US-TWENO scheme. Then, we devise a novel troubled cell indicator capable of accurately identifying troubled cells containing strong discontinuities: the existence of extreme points of the trigonometric polynomials within the smallest interval (the target cell itself) is determined by whether the estimated minimum and maximum values of their derivative trigonometric polynomials have opposite signs. To the best of our knowledge, this is the first troubled cell indicator devised specifically within the target cell interval. The HUS-TWENO scheme is improved, offering greater efficiency, lower dissipation, and higher resolution. Numerical experiments demonstrate the effectiveness of the HUS-TWENO scheme.

1. Introduction

Partial differential equations, particularly hyperbolic conservation laws, are fundamental to modeling practical problems in aviation, oceanography, and water conservancy [1,2]. If lower-order numerical approaches are employed to handle hyperbolic conservation laws, the numerical solution tends to exhibit considerable dissipation around the areas containing the discontinuities. Therefore, it is essential to construct a high-order essentially non-oscillatory (ENO) scheme for addressing hyperbolic conservation laws, since the numerical solution has high precision in smooth areas while maintaining an ENO behavior near strong discontinuities. In recent years, researchers have developed numerous advanced methods, such as the weighted ENO (WENO) scheme [3,4,5,6,7], discontinuous Galerkin (DG) finite element method [4,8], and weighted compact nonlinear scheme (WCNS) [9], etc. This article focuses on developing a novel fifth-order unequal-sized trigonometric WENO (US-TWENO) scheme and a novel hybrid US-TWENO (HUS-TWENO) scheme featuring an innovative troubled cell indicator within a finite difference structure for dealing with hyperbolic conservation laws on structured grids.

The initial finite volume WENO scheme was introduced by Liu et al. [10]. Subsequently, it was improved and extended to the finite difference system by Jiang and Shu [3] in 1996. Henrick et al. [11] discovered that the classical WENO-JS scheme [3] was unable to attain the optimal fifth-order accuracy at the critical points and created the WENO-M scheme by designing a mapping function to deal with this drawback. Borges et al. [12] designed the WENO-Z scheme, which not only maintained the intended level of accuracy near critical points but also decreased additional computational cost. Later, Zhu and Qiu further improved the WENO scheme by developing some unequal-sized WENO (US-WENO) schemes [13,14] in the algebraic polynomial space and adapted it on unstructured grids. The superior linear weights could be any positive values, as long as their total sum adds up to one. This made the application of current WENO schemes [3,11,12] much simpler, especially on two-dimensional and three-dimensional unstructured grids.

However, although the abovementioned high-order WENO schemes were effective in numerically addressing hyperbolic conservation laws, these advanced numerical approaches were mainly constructed by using the algebraic polynomials that could not be adapted to the characteristics of given data with significant fluctuations. For interpolating data with significant oscillations, the numerical scheme utilizing the trigonometric polynomials was more appropriate for solving such issues than the numerical scheme based on algebraic polynomials. In 1996, Christofi [15] constructed a class of local trigonometric polynomial interpolation methods and associated trigonometric ENO schemes. Then, Zhu and Qiu [16] developed trigonometric WENO (TWENO) schemes for simulating hyperbolic conservation laws and highly oscillatory problems. On this basis, Wang et al. constructed a novel fifth-order TWENO scheme [17] that incorporated the multi-resolution TWENO scheme [18] combined with multi-resolution techniques to achieve any desired level of precision in smooth areas. Although the WENO scheme had been widely used as a significant shock-capturing scheme in practical engineering problems, the local characteristic decomposition, smooth indicators, flux splitting, and complex nonlinear weight calculations all resulted in substantially additional computation costs.

The high-order WENO schemes mentioned earlier required sophisticated spatial reconstruction techniques of numerical flux splitting, local characteristic projection, and the complex and time-consuming computations of the optimal linear weights, smoothness indicators, and nonlinear weights to sustain the stability for the system. One effective option was the hybrid WENO scheme, which combined the expensive WENO reconstruction nearby discontinuous regions and used the high-order linear upwind interpolation in smooth zones, thereby optimizing the benefits of each approach [19]. A crucial element of the hybrid WENO scheme was the discontinuity indicator. At present, the existing discontinuity indicators mainly relied on the troubled cell indicators employed in the DG techniques [20]. Based on the research, Li and Qiu [21] systematically studied and evaluated the effectiveness of different troubled cell indicators for the development of hybrid WENO schemes. Furthermore, the troubled cells containing the discontinuities could also be detected by determining the locations of the polynomials’ extreme points [22]. However, among these troubled cell indicators, some of them containing artificially adjustable parameters were not easily applied in large-scale engineering applications, while the others were not easily generalized to higher-order numerical schemes. An excessively wide identification interval might result in misidentifying the troubled cells, leading to higher computation expenses. As a result, numerous researchers had focused on studying the hybrid WENO schemes, for instance, the hybrid compact-WENO scheme introduced by Pirozzoli [23], the characteristic-wise hybrid scheme introduced by Ren et al. [24] based on the conservative compact scheme [23], and the hybrid central-WENO finite difference scheme proposed by Costa and Don [25]. For further investigations into the hybrid WENO schemes, the readers can consult [26,27,28,29,30,31,32,33,34,35,36] and their references.

Within this study, we proceed with the novel fifth-order finite difference US-TWENO scheme and subsequently propose the novel troubled cell indicator to design the novel hybrid US-TWENO scheme, which combines related linear upwind reconstruction and the US-TWENO spatial reconstruction on structured grids. Following the ideas proposed in [14,16,17,18], this work introduces a novel fifth-order finite difference trigonometric WENO scheme featuring a convex combination of one quartic trigonometric polynomial and two linear trigonometric polynomials in the novel trigonometric polynomial space. The new fifth-order US-TWENO scheme could obtain smaller $L^1$ and $L^{\infty }$ errors in smooth areas and diminish to the second-order approximation in nonsmooth areas. In a general sense, the novelties of this TWENO scheme are found in some areas: an innovative TWENO-styled exchange among three different-degree trigonometric polynomials for high-precision approximations in smooth zones, the non-oscillatory behavior near shocks or contact discontinuities, and the superior treatment of the wave-like and highly oscillatory issues. After that, we propose the novel fifth-order HUS-TWENO scheme. We detect the existence of extreme points of reconstruction trigonometric polynomials within the smallest interval (the target cell) $I_i=[x_{i-1/2},x_{i+1/2}]$ determined by whether the estimated minimum and maximum values of their derivative trigonometric polynomials have opposite signs or not. When the signs are opposite, $I_i$ is established as a troubled cell and the complex US-TWENO reconstruction is used. If not, the efficient linear upwind scheme is utilized directly. The HUS-TWENO scheme has many advantages. Firstly, the troubled cell indicator is formulated from two simple trigonometric polynomials, requiring no manual parameters related to the problems. Secondly, during the identification of troubled cells, the troubled cell indicator determines if the reconstruction trigonometric polynomials contain extreme points by analyzing the varying signs in the estimated minima and maxima of their corresponding derivative trigonometric polynomials. Thus, this method is simpler and more efficient than the one in [37,38], requiring an intricate procedure to find the zero points of one-dimensional derivative algebraic polynomials across a wider interval $[x_{i-5/2},x_{i+5/2}]$, which could not be directly applied to higher dimensions. Furthermore, this troubled cell indicator acts only in the most reduced interval $[x_{i-1/2},x_{i+1/2}]$, smaller than the one in [37,38,39] in the algebraic polynomial space. As a result, it enables more precise identification of the troubled cells. Finally, the fifth-order finite difference HUS-TWENO scheme can accurately distinguish smooth regions from discontinuous ones, seamlessly transitioning between the simple linear upwind reconstruction and complex US-TWENO reconstruction, thus substantially improving the efficiency.

The structure of the remaining parts is outlined below. In Section 2, the novel fifth-order finite difference US-TWENO and HUS-TWENO schemes with the novel troubled cell indicator are introduced to address hyperbolic conservation laws, especially for some highly oscillatory problems. Section 3 presents several numerical evaluations to verify the high-order precision, high efficiency, and good performance of the novel US-TWENO and HUS-TWENO schemes, respectively. Section 4 provides some concluding remarks in the end.

2. US-TWENO and HUS-TWENO Schemes

One-dimensional hyperbolic conservation laws are

$$\left\{\begin{array}{c}{u}_t+f{\left(u\right)}_x=0,\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$$

(1)

and associated semi-discrete expression for (1) is

$${\displaystyle \frac{du}{dt}}=L\left(u\right),$$

(2)

wherein $L\left(u\right)$ represents the high-order finite difference spatial discretization of $-f_x\left(u\right)$. For clarity, the computational domain $[a,b]$ is split into uniform intervals $I_i=[x_{i-1/2},x_{i+1/2}]$ using a constant grid spacing $h={\displaystyle \frac{b-a}{N}}=x_{i+1/2}-x_{i-1/2}$ (N denotes the count of grid points). $x_i={\displaystyle \frac{1}{2}}(x_{i+1/2}+x_{i-1/2})$ is the cell center associated with the interval $I_i$. $u_i\left(t\right)$ stands for the numerical solution. $u(x_i,t)$ is the nodal point value. Following this, the right-hand side of Equation (2) is

$$L\left(u_i\left(t\right)\right)=-{\displaystyle \frac{1}{h}}({\widehat{f}}_{i+1/2}-{\widehat{f}}_{i-1/2}),$$

(3)

in which ${\widehat{f}}_{i\pm 1/2}$ denote the numerical fluxes at the interfaces $x_{i\pm 1/2}$ of the target cell $I_i$. The scheme achieves a fifth-order accuracy provided that

$${\displaystyle \frac{1}{h}}({\widehat{f}}_{i+1/2}-{\widehat{f}}_{i-1/2})=f{\left(u\right)}_x{|}_{x=x_i}+O\left(h^5\right).$$

(4)

For maintaining the scheme’s stability, $f\left(u\right)$ can be divided into two components $f^{\pm }\left(u\right)$, where ${\displaystyle \frac{d{f}^{+}\left(u\right)}{du}}\ge 0$ and ${\displaystyle \frac{d{f}^{-}\left(u\right)}{du}}\le 0$. Both exhibit smoothness as functions of u. And we call for

$$f\left(u\right)=f^{+}\left(u\right)+f^{-}\left(u\right).$$

(5)

As a popular approach, the Lax–Friedrichs flux splitting is selected by

$$f^{\pm }\left(u\right)={\displaystyle \frac{1}{2}}(f\left(u\right)\pm \alpha u),$$

(6)

in this context, $\alpha$ represents the upper limit of $|f^{\prime }\left(u\right)|$ within the range of u. Based on this, the numerical flux ${\widehat{f}}_{i+1/2}$ can be defined as

$${\widehat{f}}_{i+1/2}=f_{i+1/2}^{+}+f_{i+1/2}^{-}.$$

(7)

2.1. US-TWENO Scheme

On the premise that the cell averages ${\overline{w}}_j={\displaystyle \frac{1}{h}}{\int }_{x_{j-1/2}}^{x_{j+1/2}}w\left(x\right)dx$ for every j that is recognized, we intend to achieve a related degree trigonometric WENO polynomial approximation $w_i\left(x\right)$ limited to the target cell $I_i$, which relies on the stencils comprising $I_j=[x_{j-1/2},x_{j+1/2}]$ for $j=i-2,\dots ,i+2$. The steps can be summarized in the following. The numerical fluxes in (3) can be computed with the finite difference TWENO schemes. These characteristics provide numerous benefits to the TWENO schemes, including simplicity, efficiency, and robustness. This subsection uses the trigonometric polynomials instead of the algebraic polynomials to design the unequal-sized WENO spatial approximations for solving the highly oscillatory problems or wave-like phenomena. This US-TWENO scheme is reconstructed by using the data specified on a single five-point stencil and two two-point stencils in the new trigonometric polynomial space rather than relying on three equal-sized three-point stencils [3] in algebraic polynomial space. This approach is intended to provide high-order precision in smooth areas while maintaining its ENO characteristics close to strong discontinuities. This method allows the optimal linear weights of the US-TWENO scheme to be any positive parameters as long as their sum equals one. Additionally, it can be readily adapted for use in multi-dimensional cases. The following part provides a detailed explanation of the new high-order spatial process.

Step 1. Select one five-point stencil $T_1=\{I_{i-2},I_{i-1},I_i,I_{i+1},I_{i+2}\}$ and two two-point stencils $T_2=\{I_{i-1},I_i\}$, $T_3=\{I_i,I_{i+1}\}$ to obtain a quartic trigonometric polynomial $p_1\left(x\right)\in span\{{\phi }_0\left(x\right),{\phi }_1\left(x\right),{\phi }_2\left(x\right),{\phi }_3\left(x\right),{\phi }_4\left(x\right)\}$ and two linear trigonometric polynomials $p_2\left(x\right),p_3\left(x\right)\in span\{{\phi }_0\left(x\right),{\phi }_1\left(x\right)\}$, respectively, which satisfy

$${\displaystyle \frac{1}{h}}{\int }_{x_{j-1/2}}^{x_{j+1/2}}{p}_1\left(x\right)dx={\overline{w}}_j,j=i-2,\dots ,i+2,$$

(8)

$${\displaystyle \frac{1}{h}}{\int }_{x_{j-1/2}}^{x_{j+1/2}}{p}_2\left(x\right)dx={\overline{w}}_j,j=i-1,i,$$

(9)

$${\displaystyle \frac{1}{h}}{\int }_{x_{j-1/2}}^{x_{j+1/2}}{p}_3\left(x\right)dx={\overline{w}}_j,j=i,i+1.$$

(10)

In which ${\phi }_0\left(x\right)=1$, ${\phi }_1\left(x\right)=sin(x-x_i)$, ${\phi }_2\left(x\right)={sin}^2(x-x_i)$, ${\phi }_3\left(x\right)=sin(x-x_i)-(x-x_i)$, ${\phi }_4\left(x\right)={sin}^2(x-x_i)-{(x-x_i)}^2$. They are different to the trigonometric polynomial space that specified in [16,17,18]. Therefore, we possess

$$p_1\left(x\right)=a_0{\phi }_0\left(x\right)+a_1{\phi }_1\left(x\right)+a_2{\phi }_2\left(x\right)+a_3{\phi }_3\left(x\right)+a_4{\phi }_4\left(x\right),$$

(11)

$$p_2\left(x\right)=b_0{\phi }_0\left(x\right)+b_1{\phi }_1\left(x\right),$$

(12)

$$p_3\left(x\right)=c_0{\phi }_0\left(x\right)+c_1{\phi }_1\left(x\right),$$

(13)

where

$$\begin{array}{cc}\hfill a_0=& -{csc}^{5}h((-24h+27sinh-sin\left(3h\right)){\overline{w}}_{i-2}+(96h-98sinh-sin\left(3h\right)\hfill \\ \hfill & +sin\left(5h\right)){\overline{w}}_{i-1}+(-144h-98sinh+124sin\left(3h\right)-26sin\left(5h\right)){\overline{w}}_i+(96h\hfill \\ \hfill & -98sinh-sin\left(3h\right)+sin\left(5h\right)){\overline{w}}_{i+1}+(-24h+27sinh-sin\left(3h\right)){\overline{w}}_{i+2})/384,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill a_1=& {cos}^3(h/2){csc}^{4}h((h^2-cos(h/2)+cos(3h/2)){\overline{w}}_{i-2}\hfill \\ \hfill & -(2h^2-cos(3h/2)+cos(5h/2)){\overline{w}}_{i-1}+(2h^2-cos(3h/2)+cos(5h/2)){\overline{w}}_{i+1}\hfill \\ \hfill & +(-h^2+cos(h/2)-cos(3h/2)){\overline{w}}_{i+2})/\left(2h\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill a_2=& -{csc}^{5}h((4h^3-3sinh+sin\left(3h\right)){\overline{w}}_{i-2}+(-16h^3+2sinh+sin\left(3h\right)\hfill \\ \hfill & -sin\left(5h\right)){\overline{w}}_{i-1}+(24h^3+2sinh-4sin\left(3h\right)+2sin\left(5h\right)){\overline{w}}_i+(-16h^3\hfill \\ \hfill & +2sinh+sin\left(3h\right)-sin\left(5h\right)){\overline{w}}_{i+1}+(4h^3-3sinh+sin\left(3h\right)){\overline{w}}_{i+2})/\left(32h^2\right),\hfill \end{array}$$

(16)

$$a_3={csc}^2(h/2)({\overline{w}}_{i-2}-2cosh{\overline{w}}_{i-1}+2cosh{\overline{w}}_{i+1}-{\overline{w}}_{i+2})/\left(8h\right),$$

(17)

$$\begin{array}{cc}\hfill a_4=& {csc}^{2}h(-{\overline{w}}_{i-2}+(2+2cos\left(2h\right)){\overline{w}}_{i-1}-(2+4cos\left(2h\right)){\overline{w}}_i\hfill \\ \hfill & +(2+2cos\left(2h\right)){\overline{w}}_{i+1}-{\overline{w}}_{i+2})/\left(8h^2\right),\hfill \end{array}$$

(18)

$$b_0={\overline{w}}_i,$$

(19)

$$b_1=-{\displaystyle \frac{h}{2}}csc{\displaystyle \frac{h}{2}}csch{\displaystyle \frac{{\overline{w}}_{i-1}-{\overline{w}}_i}{2}},$$

(20)

$$c_0={\overline{w}}_i,$$

(21)

$$c_1=-{\displaystyle \frac{h}{2}}csc{\displaystyle \frac{h}{2}}csch{\displaystyle \frac{{\overline{w}}_i-{\overline{w}}_{i+1}}{2}}.$$

(22)

Step 2. The linear weights are selected arbitrarily as some positive values, which are denoted by ${\gamma }_n$, $n=1,2,3$, with their summation being one [17,36]. Because the arbitrary selection of linear weights does not affect the optimal precision, as an example, the linear weights are predetermined to be ${\gamma }_1=0.8$, ${\gamma }_2=0.1$, and ${\gamma }_3=0.1$, which are similar to that specified in [36].

Step 3. Calculate the smoothness indicators ${\beta }_n$ of the trigonometric polynomials $p_n\left(x\right)$ over the interval $I_i$ by applying the standard method in [3,5,6,14],

$${\beta }_n=\sum _{\alpha =1}^{\kappa }{\int }_{x_{i-1/2}}^{x_{i+1/2}}{h}^{2\alpha -1}{\left({\displaystyle \frac{d^{\alpha }{p}_n\left(x\right)}{d{x}^{\alpha }}}\right)}^{2}dx,$$

(23)

wherein $\kappa =4$ for $n=1$ and $\kappa =1$ for $n=2$ and $n=3$, respectively. Therefore, their specific expressions are

$$\begin{array}{cc}\hfill {\beta }_1=& h(6a_1^{2}h+6a_2^{2}h+12a_1{a}_{3}h+18a_3^{2}h+12a_2{a}_{4}h+6a_4^{2}h+4a_4^2{h}^3+24a_4(a_2\hfill \\ \hfill & +a_4)hcosh-48a_3(a_1+a_3)sin(h/2)+6a_1^{2}sinh+12a_1{a}_{3}sinh+6a_3^{2}sinh\hfill \\ \hfill & -24a_2{a}_{4}sinh-24a_4^{2}sinh-3a_2^{2}sin\left(2h\right)-6a_2{a}_{4}sin\left(2h\right)-3a_4^{2}sin\left(2h\right)\hfill \\ \hfill & +6h^4({(a_1+a_3)}^{2}h+16{(a_2+a_4)}^{2}h+{(a_1+a_3)}^{2}sinh-8{(a_2+a_4)}^{2}sin\left(2h\right))\hfill \\ \hfill & +6h^2(({(a_1+a_3)}^2+4(a_2^2+2a_2{a}_4+3a_4^2))h-({(a_1+a_3)}^2+16a_4(a_2\hfill \\ \hfill & +a_4\left)\right)sinh+2{(a_2+a_4)}^{2}sin\left(2h\right))+6h^6(({(a_1+a_3)}^2+64{(a_2+a_4)}^2)h\hfill \\ \hfill & -{(a_1+a_3)}^{2}sinh+32{(a_2+a_4)}^{2}sin\left(2h\right)\left)\right)/12,\hfill \end{array}$$

(24)

$${\beta }_2={\displaystyle \frac{b_1^{2}h(h+sinh)}{2}},$$

(25)

$${\beta }_3={\displaystyle \frac{c_1^{2}h(h+sinh)}{2}}.$$

(26)

Their expansions in Taylor series about ${\overline{w}}_i$ are

$$\begin{array}{cc}\hfill {\beta }_1=& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+h^4(-16{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+13{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2)/12+h^6(2365{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2-5200{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2\hfill \\ \hfill & -52{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\prime \prime \prime }+3124{\left({\left({\overline{w}}_i\right)}^{\prime \prime \prime }\right)}^2-8{\left({\overline{w}}_i\right)}^{″}{\left({\overline{w}}_i\right)}^{\left(4\right)}-192{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\left(5\right)})/2880+O\left(h^8\right)\hfill \\ \hfill =& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2(1+O\left(h^4\right))=O\left(h^2\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\beta }_2=& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2-h^3{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{″}-h^4(-{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+3{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2+4{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\prime \prime \prime })/12+O\left(h^5\right)\hfill \\ \hfill =& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2(1+O\left(h\right))=O\left(h^2\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\beta }_3=& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+h^3{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{″}-h^4(-{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+3{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2+4{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\prime \prime \prime })/12+O\left(h^5\right)\hfill \\ \hfill =& h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2(1+O\left(h\right))=O\left(h^2\right).\hfill \end{array}$$

(29)

It is assumed that they can be rewritten as ${\beta }_1=D(1+O\left(h^4\right)),{\beta }_{2,3}=D(1+O\left(h\right))$ with non-zero constant $D=h^2{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2$ and ${\left({\overline{w}}_i\right)}^{\prime }\ne 0$.

Step 4. Determine the nonlinear weights ${\omega }_n$ as

$${\omega }_n={\displaystyle \frac{{\overline{\omega }}_n}{{\sum }_{\ell =1}^3{\overline{\omega }}_{\ell }}},n=1,2,3,$$

(30)

by using

$$\begin{array}{cc}\hfill \tau =& {\left({\displaystyle \frac{|{\beta }_1-{\beta }_2|+|{\beta }_1-{\beta }_3|}{2}}\right)}^2\hfill \\ \hfill =& \left(\right|h^3{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{″}+h^4(-17{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+16{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2+4{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\prime \prime \prime })/12+O\left(h^5\right)|\hfill \\ \hfill & +|-h^3{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{″}+h^4(-17{\left({\left({\overline{w}}_i\right)}^{\prime }\right)}^2+16{\left({\left({\overline{w}}_i\right)}^{″}\right)}^2+4{\left({\overline{w}}_i\right)}^{\prime }{\left({\overline{w}}_i\right)}^{\prime \prime \prime })/12+O\left(h^5\right){\left|\right)}^2/4\hfill \\ \hfill =& \left\{\begin{array}{c}O\left(h^6\right),{\left({\overline{w}}_i\right)}^{\prime }\ne 0,{\left({\overline{w}}_i\right)}^{″}\ne 0,\\ O\left(h^8\right),{\left({\overline{w}}_i\right)}^{\prime }=0,{\left({\overline{w}}_i\right)}^{″}\ne 0,\end{array}\right.\hfill \end{array}$$

(31)

$${\overline{\omega }}_n={\gamma }_n\left(1+{\displaystyle \frac{\tau }{\epsilon +{\beta }_n}}\right)=\left\{\begin{array}{c}{\gamma }_n+O\left(h^4\right),{\left({\overline{w}}_i\right)}^{\prime }\ne 0,{\left({\overline{w}}_i\right)}^{″}\ne 0,\\ {\gamma }_n+O\left(h^6\right),{\left({\overline{w}}_i\right)}^{\prime }=0,{\left({\overline{w}}_i\right)}^{″}\ne 0,\end{array}\right.$$

(32)

with $\epsilon \ll {\beta }_n$, $\epsilon ={10}^{-6}$ is a tiny positive constant to avoid having a zero denominator. Therefore, the nonlinear weights ${\omega }_n(n=1,2,3)$ satisfy the accuracy condition ${\omega }_n={\gamma }_n+O\left(h^4\right)$ [12], which enables the WENO scheme to achieve formal fifth-order accuracy [3,5].

Step 5. The ultimate reconstruction polynomial $w_i\left(x\right)$ at $x_{i+1/2}$ is formulated as

$$w_i\left(x_{i+1/2}\right)={\displaystyle \frac{{\omega }_1}{{\gamma }_1}}\left(p_1\left(x_{i+1/2}\right)-\sum _{\ell =2}^3{\gamma }_{\ell }{p}_{\ell }\left(x_{i+1/2}\right)\right)+\sum _{\ell =2}^3{\omega }_{\ell }{p}_{\ell }\left(x_{i+1/2}\right).$$

(33)

$f_{i+1/2}^{+}=w_i\left(x_{i+1/2}\right)$ by setting ${\overline{w}}_j=f^{+}\left(u_j\right)$. The numerical flux $f_{i+1/2}^{-}$ is reconstructed with mirror symmetry relative to $x_{i+1/2}$. As the final step, the semi-discrete scheme (2) uses a fourth-order TVB Runge–Kutta approach [40]

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}{u}^{\left(1\right)}\hfill & =\hfill & u^n+{\displaystyle \frac{1}{2}}\mathsf{\Delta }tL\left(u^n\right),\hfill \\ u^{\left(2\right)}\hfill & =\hfill & u^n+{\displaystyle \frac{1}{2}}\mathsf{\Delta }tL\left(u^{\left(1\right)}\right),\hfill \\ u^{\left(3\right)}\hfill & =\hfill & u^n+\mathsf{\Delta }tL\left(u^{\left(2\right)}\right),\hfill \\ u^{n+1}\hfill & =\hfill & -{\displaystyle \frac{1}{3}}{u}^n+{\displaystyle \frac{1}{3}}{u}^{\left(1\right)}+{\displaystyle \frac{2}{3}}{u}^{\left(2\right)}+{\displaystyle \frac{1}{3}}{u}^{\left(3\right)}+{\displaystyle \frac{1}{6}}\mathsf{\Delta }tL\left(u^{\left(3\right)}\right),\hfill \end{array}\right.\end{array}$$

(34)

to obtain the full time discretization.

Remark 1.

We solely present the formulation of the fifth-order US-TWENO scheme in one dimension. Finally, it is easy to adapt the US-TWENO spatial reconstruction techniques for two dimensions in a dimension-by-dimension approach.

2.2. HUS-TWENO Scheme

Following the numerical simulation experiments described in the subsequent part, it is proven that the developed US-TWENO scheme exhibits excellent precision in the zones with smooth variations and outstanding shock-capturing ability near strong discontinuities. Nonetheless, this enhanced performance comes at the cost of increasing the computational time. Therefore, a novel hybrid TWENO scheme is constructed to decrease the processing time based on the US-TWENO scheme mentioned above. The hybrid US-TWENO scheme strategically employs the high-order linear upwind scheme in smooth zones, which are the majority of the calculation area and the US-TWENO scheme close to strong discontinuities. Therefore, under the same conditions, the hybrid US-TWENO scheme has lower errors than the US-TWENO scheme. For the purpose of accurately, automatically, and efficiently identifying the troubled cells containing strong discontinuities, a novel troubled cell indicator is introduced through an application of the trigonometric polynomials. Therefore, the novel hybrid US-TWENO scheme for dealing with hyperbolic conservation laws is introduced in this subsection. The details are provided in the following.

Step 1. Obtain the first derivative of the quartic trigonometric polynomial $p_1\left(x\right)$ as

$$p_1^{\prime }\left(x\right)=a_1{\phi }_1^{\prime }\left(x\right)+a_2{\phi }_2^{\prime }\left(x\right)+a_3{\phi }_3^{\prime }\left(x\right)+a_4{\phi }_4^{\prime }\left(x\right),$$

(35)

where ${\phi }_1^{\prime }\left(x\right)=cos(x-x_i),{\phi }_2^{\prime }\left(x\right)=sin\left(2(x-x_i)\right),{\phi }_3^{\prime }\left(x\right)=cos(x-x_i)-1,{\phi }_4^{\prime }\left(x\right)=sin\left(2(x-x_i)\right)-2(x-x_i)$.

Step 2. Identify the extreme points of the reconstruction polynomial $p_1\left(x\right)$. Determine whether the interval $I_i$ is a troubled cell containing strong discontinuities or not, which is based on whether the cubic trigonometric polynomial $p_1\left(x\right)$ has at least one extreme point within it (that is, based on whether $p_1^{\prime }\left(x\right)$ has at least one zero point). This is because if polynomial $p_1\left(x\right)$ has extremum points in the interval $I_i$, there may be discontinuities in this interval. Although [38] firstly determined the troubled cells by exactly solving the specific expressions of the zero points for the cubic algebraic polynomials in the range $[x_{i-5/2},x_{i+5/2}]$ in [36,38], it is hard to solve the zero points for the cubic trigonometric polynomials within $[x_{i-1/2},x_{i+1/2}]$, which is more narrow than that specified in [18,36,37,38]. Therefore, this article only judges whether the minimum and maximum values of the cubic trigonometric polynomial $p_1^{\prime }\left(x\right)$ within $[x_{i-1/2},x_{i+1/2}]$ have different signs to determine whether the cubic trigonometric polynomial $p_1^{\prime }\left(x\right)$ has the zero points within $[x_{i-1/2},x_{i+1/2}]$ or not.

Step 3. Determine the minimum and maximum values of the cubic trigonometric polynomial $p_1^{\prime }\left(x\right)$ within $[x_{i-1/2},x_{i+1/2}]$. This technology is still quite difficult for the high-degree trigonometric polynomials. However, through observation, it can be observed that each term $a_1{\phi }_1^{\prime }\left(x\right),a_2{\phi }_2^{\prime }\left(x\right),a_3{\phi }_3^{\prime }\left(x\right),a_4{\phi }_4^{\prime }\left(x\right)$ of the cubic trigonometric polynomial $p_1^{\prime }\left(x\right)$ shows a strictly monotonic behavior within very small intervals $[x_{i-1/2},x_i]$ and $[x_i,x_{i+1/2}]$ based on ${\phi }_1^{″}\left(x\right)=-sin(x-x_i),{\phi }_2^{″}\left(x\right)=2cos\left(2(x-x_i)\right),{\phi }_3^{″}\left(x\right)=-sin(x-x_i),{\phi }_4^{″}\left(x\right)=2cos\left(2(x-x_i)\right)-2$ and (15)–(18). This implies that the maximum and minimum values of each term occur at the endpoints of both smaller intervals. As a result, it is not necessary to obtain a specific expression for the extreme point to roughly determine its position based on this property. On this basis, we calculate the maximum value $M_1$ and minimum value $m_1$ within $[x_{i-1/2},x_i]$, and the maximum value $M_2$ and minimum value $m_2$ within $[x_i,x_{i+1/2}]$, respectively.

$$\begin{array}{cc}\hfill M_1=& max\left(a_1{\phi }_1^{\prime }\left(x_{i-1/2}\right),a_1{\phi }_1^{\prime }\left(x_i\right)\right)+max\left(a_2{\phi }_2^{\prime }\left(x_{i-1/2}\right),a_2{\phi }_2^{\prime }\left(x_i\right)\right)\hfill \\ \hfill & +max\left(a_3{\phi }_3^{\prime }\left(x_{i-1/2}\right),a_3{\phi }_3^{\prime }\left(x_i\right)\right)+max\left(a_4{\phi }_4^{\prime }\left(x_{i-1/2}\right),a_4{\phi }_4^{\prime }\left(x_i\right)\right),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill m_1=& min\left(a_1{\phi }_1^{\prime }\left(x_{i-1/2}\right),a_1{\phi }_1^{\prime }\left(x_i\right)\right)+min\left(a_2{\phi }_2^{\prime }\left(x_{i-1/2}\right),a_2{\phi }_2^{\prime }\left(x_i\right)\right)\hfill \\ \hfill & +min\left(a_3{\phi }_3^{\prime }\left(x_{i-1/2}\right),a_3{\phi }_3^{\prime }\left(x_i\right)\right)+min\left(a_4{\phi }_4^{\prime }\left(x_{i-1/2}\right),a_4{\phi }_4^{\prime }\left(x_i\right)\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill M_2=& max\left(a_1{\phi }_1^{\prime }\left(x_i\right),a_1{\phi }_1^{\prime }\left(x_{i+1/2}\right)\right)+max\left(a_2{\phi }_2^{\prime }\left(x_i\right),a_2{\phi }_2^{\prime }\left(x_{i+1/2}\right)\right)\hfill \\ \hfill & +max\left(a_3{\phi }_3^{\prime }\left(x_i\right),a_3{\phi }_3^{\prime }\left(x_{i+1/2}\right)\right)+max\left(a_4{\phi }_4^{\prime }\left(x_i\right),a_4{\phi }_4^{\prime }\left(x_{i+1/2}\right)\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill m_2=& min\left(a_1{\phi }_1^{\prime }\left(x_i\right),a_1{\phi }_1^{\prime }\left(x_{i+1/2}\right)\right)+min\left(a_2{\phi }_2^{\prime }\left(x_i\right),a_2{\phi }_2^{\prime }\left(x_{i+1/2}\right)\right)\hfill \\ \hfill & +min\left(a_3{\phi }_3^{\prime }\left(x_i\right),a_3{\phi }_3^{\prime }\left(x_{i+1/2}\right)\right)+min\left(a_4{\phi }_4^{\prime }\left(x_i\right),a_4{\phi }_4^{\prime }\left(x_{i+1/2}\right)\right).\hfill \end{array}$$

(39)

Step 4. According to (36)–(39), if $(M_1\pm h)·(m_1\pm h)<0$ or $(M_2\pm h)·(m_2\pm h)<0$, the interval (the target cell) $I_i=[x_{i-1/2},x_{i+1/2}]$ is determined to be a troubled cell and the nonlinear US-TWENO scheme is used for obtaining the fifth-order spatial approximation. Otherwise, it is not recognized as a troubled cell and the fifth-order linear upwind scheme is employed for obtaining the same order spatial approximation. Here, h represents the spatial grid step size, used to filter out grid points where the maximum and minimum values are nearly identical.

Finally, for the convenience of the readers, the proposed algorithm for the one-dimensional case can be summarized as Algorithms 1 and 2 below.

Algorithm 1 The proposed troubled cell indicator

Input: Density $\rho$ Output:  $Fla{g}_i$ Compute the coefficients $a_{0,\dots ,4}$ of polynomial $p_1^{\prime }\left(x\right)$; for $i=0$ to N do           Compute $M_1$, $m_1$, $M_2$ and $m_2$ by Equations (36)–(39)      if $(M_1\pm h)·(m_1\pm h)<0$ or $(M_2\pm h)·(m_2\pm h)<0$ then           $Fla{g}_i=1$     else           $Fla{g}_i=0$     end end return  $Fla{g}_i$

Algorithm 2 HUS-TWENO scheme

Input: $Fla{g}_i$ and $f^{+}\left(u_j\right)(j=i-2,\dots ,i+2)$ Output:  $f_{i+1/2}^{+}$ if  $Fla{g}_i=0$  then       $f_{i+1/2}^{+}=p_1\left(x_{i+1/2}\right)$ obtained by the linear method; else       $f_{i+1/2}^{+}=w_i\left(x_{i+1/2}\right)$ obtained by the US-TWENO scheme described in Section 2.1; end return  $f_{i+1/2}^{+}$

Remark 2.

In this article, the fluxes in (11) are substituted by the density ρ for detecting the troubled cells. Alternatively, the fluxes can be directly used and result in a different count of the troubled cells without impacting other algorithm properties. It is also worth noting that the identification algorithm for designing the troubled cells is manual parameter-free related to the problems, which makes it convenient for simulating some large-scale engineering problems.

Remark 3.

In order to reduce spurious oscillations when simulating multi-dimensional Euler equations, the US-TWENO scheme is applied along the local characteristic directions [3,5]. The identification of the troubled cells and the high-order linear upwind schemes are still executed in the physical space to save time, since their implementation by using the characteristic variables, computing the optimal linear weights, computing the complicated smoothness indicators, and computing the nonlinear weights are lengthy. Obviously, the HUS-TWENO scheme achieves formal fifth-order accuracy, since the analysis above has shown that the US-TWENO scheme achieves formal fifth-order accuracy, and the linear upwind scheme is of fifth-order accuracy.

3. Numerical Tests

This part presents the numerical simulations and overall CPU time for the HUS-TWENO scheme and makes a comparison with that of the US-TWENO scheme. Reference [17] has demonstrated that the WENO scheme constructed using trigonometric polynomials performs better in simulating highly oscillatory problems than algebraic polynomials. This article will not elaborate further. The CFL value is 0.6. For simplicity, the TWENO scheme [17] is also referred to as the TWENO scheme in the simulations. ${\gamma }_1=0.8$, ${\gamma }_2=0.1$, and ${\gamma }_3=0.1$ are set as examples for all numerical experiments.

Table 1. CPU time and ratio of the HUS-TWENO scheme and US-TWENO scheme from Example 1 to Example 8.

Numerical Examples	HUS-TWENO (s)	US-TWENO (s)	Ratio
1 1D-Burgers’ equation	3.67187	8.14062	0.45:1
2 2D-Burgers’ equation	462.218	996.250	0.46:1
3 $\rho (x,0)=1+0.99sin\left(x\right)$	18.1719	67.2344	0.27:1
3 $\rho (x,0)=1+0.999sin\left(x\right)$	54.5313	200.188	0.27:1
4 $\rho (x,y,0)=1+0.99sin(x+y)$	6851.39	22,298.9	0.30:1
4 $\rho (x,y,0)=1+0.999sin(x+y)$	20,256.0	64,044.9	0.31:1
5 Shu-Osher problem	0.39063	0.70313	0.55:1
6 Shock/turbulence problem	1.39063	2.78125	0.50:1
7 The advection problem	66.8906	104.844	0.63:1
8 Riemann problem with (48)	2030.30	3755.67	0.54:1
8 Riemann problem with (49)	1901.77	3543.98	0.53:1
9 Double Mach reflection	670,540	1,512,989	0.44:1

shows the overall CPU time and ratio of the HUS-TWENO and US-TWENO schemes for every test. In comparison to the US-TWENO scheme, the HUS-TWENO scheme can approximately reduce 37–73% CPU time for one-dimensional and two-dimensional numerical tests.

Experimental Environment and Configuration: All experiments were conducted on a desktop computer equipped with an Intel Core i7-13700K CPU (16 cores), running the Microsoft Windows 10 operating system. The code was compiled using the Intel Visual Fortran compiler. Timing was performed targeting the core computational portion of each algorithm. We executed 10 independent runs, discarding the first run (as a warm-up), and reported the arithmetic mean of the remaining 9 run times.

Example 1.

1D-Burgers’ equation

$$\left\{\begin{array}{c}{u}_t+{\left({\displaystyle \frac{u^2}{2}}\right)}_x=0,0<x<2,\hfill \\ u(x,0)=u_0\left(x\right),\hfill \end{array}\right.$$

(40)

wherein $u(x,0)=0.5+sin\left(\pi x\right)$ and the periodic boundary constraints.

Table 2. 1D-Burgers’ equation. Initial condition $u(x,0)=0.5+sin\left(\pi x\right)$. $T=0.5/\pi$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
40	1.83E-4		9.65E-4		1.84E-4		9.77E-4
80	4.16E-6	5.46	4.96E-5	4.28	4.18E-6	5.46	5.00E-5	4.29
160	1.25E-7	5.06	1.62E-6	4.94	1.25E-7	5.06	1.63E-6	4.94
320	3.82E-9	5.03	5.02E-8	5.01	3.84E-9	5.03	5.06E-8	5.01
640	1.18E-10	5.02	1.54E-9	5.03	1.18E-10	5.02	1.55E-9	5.03
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
40	3.50E-5				4.10E-4
80	8.10E-7		5.43		1.19E-5		5.11
160	1.66E-8		5.61		2.48E-7		5.58
320	3.56E-10		5.54		5.15E-9		5.59
640	8.54E-12		5.38		1.20E-10		5.42

displays the errors and numerical orders for the TWENO, US-TWENO and HUS-TWENO schemes at $t=0.5/\pi$. It illustrates that the HUS-TWENO scheme produces smaller numerical errors compared with the other two schemes on the same meshes, and the intended numerical orders of accuracy are attained on finer meshes. In Figure 1, we use these three schemes to show the relationship between the numerical errors and CPU time. It shows that the HUS-TWENO scheme is more efficient in terms of CPU time. Table 1 shows that the HUS-TWENO scheme reduces nearly 55% CPU time compared to the US-TWENO scheme in this example. Therefore, the HUS-TWENO scheme demonstrates higher efficiency than the US-TWENO scheme.

Example 2.

2D-Burgers’ equation

$$\left\{\begin{array}{c}{u}_t+{\left({\displaystyle \frac{u^2}{2}}\right)}_x+{\left({\displaystyle \frac{u^2}{2}}\right)}_y=0,0<x,y<4,\hfill \\ u(x,y,0)=u_0(x,y),\hfill \end{array}\right.$$

(41)

in witch $u(x,y,0)=0.5+sin\left(\pi \right(x+y)/2)$ and the periodic boundary constraints in both directions.

Table 3. 2D-Burgers’ equation. Initial condition $u(x,y,0)=0.5+sin\left(\pi \right(x+y)/2)$. $T=0.5/\pi$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
10 × 10	5.44E-2		1.23E-1		5.35E-2		1.22E-1
20 × 20	3.12E-3	4.13	9.27E-3	3.73	3.10E-3	4.11	8.91E-3	3.78
40 × 40	1.13E-4	4.79	9.21E-4	3.33	1.15E-4	4.75	9.67E-4	3.20
80 × 80	3.91E-6	4.85	4.81E-5	4.26	3.99E-6	4.86	4.97E-5	4.28
160 × 160	1.23E-7	4.99	1.57E-6	4.93	1.26E-7	4.98	1.62E-6	4.93
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
10 × 10	1.56E-2				5.82E-2
20 × 20	9.22E-4		4.08		6.68E-3		3.12
40 × 40	3.58E-5		4.68		4.11E-4		4.02
80 × 80	8.28E-7		5.44		1.20E-5		5.10
160 × 160	1.68E-8		5.62		2.51E-7		5.57

displays the errors and numerical orders for the TWENO, US-TWENO and HUS-TWENO schemes at $t=0.5/\pi$. This indicates that the HUS-TWENO scheme produces smaller numerical errors compared with the other two schemes on the same meshes, and the anticipated numerical orders of accuracy are obtained on finer meshes. In Figure 2, we use these three schemes to show the relationship between the numerical errors and CPU time. It also shows that the HUS-TWENO scheme requires less CPU time on identical grids. Table 1 shows that the HUS-TWENO scheme saves approximately 54% CPU time compared to the US-TWENO scheme. Therefore, the HUS-TWENO scheme demonstrates superior efficiency in comparison to the US-TWENO scheme when simulating this Burgers’ equation.

Example 3.

1D-Euler equations

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

(42)

ρ represents the density, u stands for the velocity along the $x-$ axis, E stands for the total energy, and p indicates the pressure. The beginning states are (1) $\rho (x,0)=1+0.99sin\left(x\right)$; (2) $\rho (x,0)=1+0.999sin\left(x\right)$; with $u(x,0)=1$, $p(x,0)=1$, $\gamma =1.4$, and $x\in [0,2\pi ]$. The periodic boundary constraints are used. The exact density includes (1) $\rho (x,t)=1+0.99sin(x-t)$; and (2) $\rho (x,t)=1+0.999sin(x-t)$. The errors and numerical accuracy levels in the density computed at $t=0.1$ using the TWENO, US-TWENO and HUS-TWENO schemes are showcased in

Table 4. 1D-Euler equations. Initial conditions $\rho (x,0)=1+0.99sin\left(x\right)$, $u(x,0)=1$, and $p(x,0)=1$. $T=0.1$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
40	7.85E-4		5.05E-3		7.83E-4		5.03E-3
80	9.92E-6	6.31	9.10E-5	5.79	9.88E-6	6.31	9.06E-5	5.79
160	5.52E-8	7.49	1.24E-6	6.20	5.52E-8	7.48	1.24E-6	6.19
320	1.81E-10	8.25	6.32E-9	7.62	1.81E-10	8.25	6.32E-9	7.61
640	8.48E-13	7.74	3.26E-11	7.60	8.48E-13	7.74	3.26E-11	7.60
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
40	4.48E-9				7.02E-9
80	5.82E-11		6.27		9.10E-11		6.27
160	8.49E-13		6.10		1.33E-12		6.10
320	1.30E-14		6.03		2.04E-14		6.03
640	2.02E-16		6.01		3.17E-16		6.01

and

Table 5. 1D-Euler equations. Initial conditions $\rho (x,0)=1+0.999sin\left(x\right)$, $u(x,0)=1$, and $p(x,0)=1$. $T=0.1$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
40	1.22E-3		7.48E-3		1.22E-3		7.45E-3
80	7.63E-5	4.00	6.12E-4	3.61	7.57E-5	4.01	6.08E-4	3.62
160	2.52E-6	4.92	5.02E-5	3.61	2.52E-6	4.91	5.02E-5	3.60
320	1.01E-8	7.96	2.94E-7	7.42	1.01E-8	7.96	2.94E-7	7.42
640	4.98E-11	7.66	2.42E-9	6.92	4.98E-11	7.66	2.42E-9	6.92
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
40	7.45E-9				1.18E-8
80	8.51E-11		6.45		1.34E-10		6.46
160	9.99E-13		6.41		1.57E-12		6.42
320	1.38E-14		6.18		2.16E-14		6.18
640	2.06E-16		6.06		3.24E-16		6.06

. Both tables show that the HUS-TWENO scheme produces smaller numerical errors compared with the other two schemes on the same meshes. Generally speaking, the desired level of precision can be obtained on finer meshes. As the meshes become much finer, we find that these three schemes achieve the anticipated numerical orders of accuracy in case (1) and case (2) in Table 4 and Table 5. In Figure 3 and Figure 4, we use these three schemes to show the association between the numerical errors and CPU time for cases (1) and (2), respectively. It shows that the HUS-TWENO scheme requires less CPU time and produces smaller numerical errors on identical grids. Table 1 shows that the HUS-TWENO scheme saves nearly 73% CPU time on different initial conditions. Therefore, the HUS-TWENO scheme demonstrates good efficiency compared with that of the US-TWENO scheme when simulating this benchmark test scenario with varying starting conditions.

Example 4.

2D-Euler equations

$${\displaystyle \frac{\partial }{\partial t}}\left(\begin{array}{c}\rho \\ \rho u\\ \rho v\\ E\end{array}\right)+{\displaystyle \frac{\partial }{\partial x}}\left(\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ u(E+p)\end{array}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(\begin{array}{c}\rho v\\ \rho uv\\ \rho v^2+p\\ v(E+p)\end{array}\right)=0.$$

(43)

ρ represents the density, u and v stand for the velocity components in the $x$-direction and the $y$-direction, respectively, E denotes the total energy, and p stands for the pressure. The starting states are (1) $\rho (x,y,0)=1+0.99sin(x+y)$; (2) $\rho (x,y,0)=1+0.999sin(x+y)$; with $u(x,y,0)=1$, $v(x,y,0)=1$, $p(x,y,0)=1$, and $\gamma =1.4$. The exact solutions for the density include (1) $\rho (x,y,t)=1+0.99sin(x+y-2t)$; and (2) $\rho (x,y,t)=1+0.999sin(x+y-2t)$, respectively.

Table 6. 2D-Euler equations. Initial conditions $\rho (x,y,0)=1+0.99sin(x+y)$, $u(x,y,0)=1$, $v(x,y,0)=1$, and $p(x,y,0)=1$. $T=0.1$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
20 × 20	4.57E-3		1.59E-2		4.50E-3		1.58E-2
40 × 40	3.01E-4	3.93	1.94E-3	3.03	2.97E-4	3.92	1.92E-3	3.04
80 × 80	5.37E-6	5.81	6.06E-5	5.00	5.35E-6	5.79	6.04E-5	4.99
160 × 160	1.68E-8	8.32	2.71E-7	7.81	1.68E-8	8.32	2.71E-7	7.80
320 × 320	6.11E-11	8.10	1.69E-9	7.32	6.11E-11	8.10	1.69E-9	7.32
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
20 × 20	6.91E-7				1.15E-6
40 × 40	8.51E-9		6.34		1.37E-8		6.38
80 × 80	1.15E-10		6.21		1.80E-10		6.25
160 × 160	1.69E-12		6.08		2.67E-12		6.07
320 × 320	2.60E-14		6.02		4.10E-14		6.03

and

Table 7. 2D-Euler equations. Initial conditions $\rho (x,y,0)=1+0.999sin(x+y)$, $u(x,y,0)=1$, $v(x,y,0)=1$, and $p(x,y,0)=1$. $T=0.1$. $L^1$ and $L^{\infty }$ errors.

	TWENO scheme				US-TWENO scheme
grid points	$L^1$ error	order	$L^{\infty }$ error	order	$L^1$ error	order	$L^{\infty }$ error	order
20 × 20	6.73E-3		2.06E-2		6.62E-3		2.03E-2
40 × 40	7.82E-4	3.11	4.14E-3	2.32	7.68E-4	3.11	4.06E-3	2.33
80 × 80	8.96E-5	3.12	8.76E-4	2.24	8.92E-5	3.11	8.73E-4	2.22
160 × 160	1.35E-6	6.05	1.94E-5	5.49	1.35E-6	6.05	1.94E-5	5.49
320 × 320	6.26E-9	7.76	1.25E-7	7.28	6.25E-9	7.75	1.25E-7	7.28
	HUS-TWENO scheme
grid points	$L^1$ error		order		$L^{\infty }$ error		order
20 × 20	1.16E-6				1.81E-6
40 × 40	1.46E-8		6.32		2.52E-8		6.17
80 × 80	1.63E-10		6.49		2.99E-10		6.40
160 × 160	2.01E-12		6.34		3.44E-12		6.44
320 × 320	2.77E-14		6.18		4.36E-14		6.30

show the errors and numerical orders of accuracy of the density calculated using the periodic boundary constraints in $[0,2\pi ]\times [0,2\pi ]$ at $t=0.1$ by the TWENO, US-TWENO and HUS-TWENO schemes. It is apparent that the HUS-TWENO scheme is able to attain smaller truncation errors compared with the other two schemes on identical meshes. And these three schemes are able to realize the desired numerical orders of accuracy. In Figure 5 and Figure 6, we use these three schemes to show the connection between the numerical errors and CPU time for cases (1) and (2), respectively. It also shows that the HUS-TWENO scheme requires less CPU time and produces lower computational errors on the identical grids. Table 1 shows that the HUS-TWENO scheme reduces about 70% CPU time in contrast to that of the US-TWENO scheme. Therefore, the HUS-TWENO scheme demonstrates better efficiency than that of the US-TWENO scheme when simulating this benchmark test case.

Example 5.

1D Shu-Osher problem [5] with initial conditions

$${(\rho ,u,p,\gamma )}^T=\left\{\begin{array}{cc}{(3.857143,2.629369,10.333333,1.4)}^T,\hfill & x\in [-5,-4),\hfill \\ {(1+0.2sin\left(5x\right),0,1,1.4)}^T,\hfill & x\in [-4,5].\hfill \end{array}\right.$$

(44)

This scenario exemplifies the interplay between a Mach 3 shock wave and a sinusoidal variation in the density. Figure 7 gives the calculated density at $t=1.8$. The reference solution is calculated by the WENO-JS scheme [3] on 2000 points. Figure 7 also displays an enlarged view of the various TWENO schemes and the temporal outcomes on the designated grid points. It indicates that the HUS-TWENO scheme outperforms the US-TWENO scheme on identical meshes due to the use of the linear upwind scheme in smooth regions. Additionally, the discontinuity indicator can automatically detect the discontinuous features without the need for any manual parameter adjustments related to the problems. And Table 1 illustrates that the HUS-TWENO scheme requires less CPU time than before.

Example 6.

A variation in the shock/turbulence issue [3,41]. It describes (42) with the initial conditions

$${(\rho ,u,p)}^T=\left\{\begin{array}{cc}{(1.515695,0.523346,1.80500)}^T,\hfill & -5<x<-4.5,\hfill \\ {(1+0.1sin\left(10\pi x\right),0.0,1.0)}^T,\hfill & -4.5<x<5,\hfill \end{array}\right.$$

(45)

which involves a right shock wave impacting a high-frequency density disturbance at a Mach number 1.1. As the shock wave propagates, the disturbance extends in the opposite direction. We calculate the flow rate at $t=5$, which exceeds the flow rate for this problem in [41] by a factor of over 10 times. The solution exhibits physical oscillations and must be solved numerically. Figure 8 displays the benchmark solution, which is a convergent result achieved through the application of the WENO-JS scheme [3] on 20,000 grid points. It also shows the density computed by the HUS-TWENO scheme and US-TWENO scheme on 700 grid points. The HUS-TWENO scheme produces better numerical results in comparison to that of the US-TWENO scheme. Moreover, the discontinuity indicator effectively detects the discontinuous structures automatically without the need for manual parameter adjustments related to the problems. And the HUS-TWENO scheme takes less CPU time compared to the US-TWENO scheme in Table 1.

Example 7.

2D advection problem with the control equation

$$u_t+{(-(y-y_0)\omega u)}_x+{\left((x-x_0)\omega u\right)}_y=0.$$

(46)

The exact solution includes the rotation of the initial value around $(x_0,y_0)=(0,0)$ and the angular velocity is assigned with $\omega =1$. The initial conditions are

$$u(x,y,0)=\left\{\begin{array}{cc}10-{\displaystyle \frac{10}{0.3}}\sqrt{{(x-0.5)}^2+y^2},\hfill & \left\{(x,y)\right|\sqrt{{(x-0.5)}^2+y^2}\le 0.3\}/\hfill \\ \hfill & \left\{\right(x,y\left)\right|y\le 0.1\cap |x-0.5|\le 0.06\},\hfill \\ 0,\hfill & otherwise(x,y)\in [-1,1]\times [-1,1].\hfill \end{array}\right.$$

(47)

This test case poses a significant computational challenge due to its high rotational angular velocity and its initial conditions, including a continuous area, local peaks and troughs, and abrupt jumps. Figure 9 illustrates the graph of the calculated variable u at $t=10\pi$. Figure 9 shows that the simulation effect computed by the HUS-TWENO scheme demonstrates a marginally superior simulation performance in comparison to that of the US-TWENO scheme on an equivalent grid level due to the use of the linear upwind scheme in smooth regions. As shown in Figure 9d, the discontinuity indicator is capable of autonomously detecting the intricate shock wave structure of the problem without adjusting any manual parameters related to the problems. Again, the HUS-TWENO scheme consumes less CPU time than the US-TWENO scheme does in Table 1.

Example 8.

2D-Euler equations for the Riemann problem [42]. It solves (43) in the domain of $[0,1]\times [0,1]$ with starting conditions

$${\left(\begin{array}{c}\rho ,u,v,p\hfill \end{array}\right)}^T=\left\{\begin{array}{cc}{(0.5313,0,0,0.4)}^T,\hfill & x>0.5,y>0.5,\hfill \\ {(1,0.7276,0,1)}^T,\hfill & x<0.5,y>0.5,\hfill \\ {(0.8,0,0,1)}^T,\hfill & x<0.5,y<0.5,\hfill \\ {(1,0,0.7276,1)}^T,\hfill & x>0.5,y<0.5,\hfill \end{array}\right.$$

(48)

and

$${\left(\begin{array}{c}\rho ,u,v,p\hfill \end{array}\right)}^T=\left\{\begin{array}{cc}{(1,0.1,0,1)}^T,\hfill & x>0.5,y>0.5,\hfill \\ {(0.5313,0.8276,0,0.4)}^T,\hfill & x<0.5,y>0.5,\hfill \\ {(0.8,0.1,0,0.4)}^T,\hfill & x<0.5,y<0.5,\hfill \\ {(0.5313,0.1,0.7276,0.4)}^T,\hfill & x>0.5,y<0.5,\hfill \end{array}\right.$$

(49)

respectively. The density results are illustrated in Figure 10 and Figure 11 at $t=0.25$ with (48) and at $t=0.3$ with (49), respectively. We can see that the HUS-TWENO scheme and US-TWENO scheme have similar simulation effects on the same grids. In addition, it can be seen in Figure 12 that the discontinuity indicator is capable of autonomously detecting the intricate shock wave structure of the problem without adjusting any manual parameters related to the problems. Once again, the HUS-TWENO scheme consumes less CPU time than the US-TWENO scheme does in Table 1.

Example 9.

Double Mach reflection problem [3]. It solves (43) in $[0,4]\times [0,1]$. The reflective partition is situated along the lower boundary of the area, beginning at the coordinates $x=16$ and $y=0$, at a 60° angle to the x-axis. The boundary conditions can be referred to in [3]. At the top boundary, there exists a Mach 10 shock wave and $\gamma =1.4$. Figure 13 shows the outcomes within the area bounded by the domain of $[0,3]\times [0,1]$ at $t=0.2$. From Figure 13, the HUS-TWENO scheme demonstrates the superior performance and reduced numerical dissipation in comparison to the US-TWENO scheme when applying on an equivalent mesh. Furthermore, it is noticeable in Figure 13c that the discontinuity indicator is capable of autonomously detecting the intricate shock wave patterns of the issue without adjusting any manual parameters related to the problems. Once again, the HUS-TWENO scheme consumes less CPU time than the US-TWENO scheme does in Table 1.

4. Concluding Remarks

This study develops a novel fifth-order finite difference US-TWENO scheme and a novel hybrid US-TWENO (HUS-TWENO) scheme with a novel troubled cell indicator. Firstly, a new finite difference US-TWENO scheme has been crafted within the new trigonometric polynomial space to address hyperbolic conservation laws and highly oscillatory issues in comparison to that specified in [16,17,18]. We could randomly set the linear weights, compute the smoothness indicators, and compute the novel nonlinear weights. The main advantages of this US-TWENO scheme lie in its robustness in solving problems containing strong shocks or high oscillations, its ability to gain the optimal fifth-order precision, and its ease of coding in programs compared to before. After that, the fifth-order finite difference HUS-TWENO scheme is developed. Among which, the novel troubled cell indicator, which requires no manual parameters related to the problems, is based on two quartic trigonometric polynomials. It detects the discontinuities by detecting the extreme points of the related reconstruction trigonometric polynomials within the narrowest interval $[x_{i-1/2},x_{i+1/2}]$, which is more constricted than $[x_{i-3/2},x_{i+3/2}]$ and $[x_{i-5/2},x_{i+5/2}]$ in [37,38] on structured grids and the two-dimensional interval in [39] on triangular grids, respectively, in the algebraic polynomial space. Moreover, we use the signs of the estimated minimum and maximum values to identify the zero points of the corresponding derivative trigonometric polynomials, avoiding complicated and time-consuming root-solving procedures. Therefore, the HUS-TWENO scheme, built using this novel troubled cell indicator, not only retains the excellent properties of the US-WENO scheme [14], whose reconstruction polynomials are in the algebraic polynomial space, but also features several distinct advantages, including the narrowest interval for detecting troubled cells, the seamless switching between the complex US-TWENO scheme and the linear upwind scheme, efficient computation, and the sharp shock transitions around strong discontinuities. Lastly, several numerical tests are conducted to demonstrate the effectiveness of the fifth-order HUS-TWENO scheme, since it can approximately reduce 37–73% CPU time relative to the novel fifth-order US-TWENO scheme on structured grids.