A Fourth-Order Symmetric WENO Scheme with Improved Performance by New Linear and Nonlinear Optimizations

Abstract

A fourth-order, symmetric, weighted essentially non-oscillatory scheme is proposed with improved linear and nonlinear properties. In order to improve its linear property as the dispersion and dissipation relations, an optimization method is developed by solving a set of equations proposed in the paper. Using this approach, optimization objectives carefully chosen are realized, through which improved linear performance is attained while maintaining stability in the shock-reflection problem. Implementation of a nonlinear scheme is another important issue which usually affects its practical performance in applications. Optimization in this regard is thought to make the scheme work in its linear form effectively and problem-independently when the flow varies relatively smoothly. To fulfill the objectives, four topics were investigated. The first is a new hybrid indicator of smoothness which is based on the concept of total variation, through which good resolution is obtained for resolving structures with short wavelength. The second is a modification specially designed for the most downwind indicator to avoid numerical oscillations. The third is a new transition algorithm to make the scheme work between its linear and nonlinear states by a variable $p_{wr}$. The new algorithm is so designed as to avoid misjudgment of smooth and oscillatory flow field. The fourth one regards case generality or problem-independence. To address four issues, the so-called rescale functions are presented separately. They are then integrated as one function for easy implementation. Using this integrated function, a fourth-order scheme for solving the Euler/Navier-Stokes equations can work in its optimized linear form in the smooth region and behave nonlinearly at discontinuities to ensure essentially oscillation-free solutions. Numerical examples manifest its capabilities to resolve waves from acoustics to shocks, flows from subsonic to hypersonic speed, and flow patterns from laminar to turbulent.

Appendices

Appendix 1

Following the analysis of Ref. [3], the necessary and sufficient condition for nonlinear weighted scheme to achieve fourth-order convergence at smooth region are first derived. Considering Eq. (3) and $r=r^{\prime }=3$, the following can be obtained for a fourth-order nonlinear scheme

$$\begin{aligned} \hat{f}_{j\pm 1/2}= & {} \sum _{k=0}^3 {C_k q_{k,j\pm 1/2}^3 } +\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) q_{k,j\pm 1/2}^3 } \\= & {} \left[ {H\left( {x_{j\pm 1/2} } \right) +B_{j\pm 1/2} \Delta x^{4}+O\left( {\Delta x^{5}} \right) } \right] +\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) q_{k,j\pm 1/2}^3 }, \end{aligned}$$

where h(x) satisfies $f(x)=\frac{1}{\Delta x}\int _{x-{\Lambda x}/2}^{x+{\Lambda x}/2} {h\left( {{x}^{\prime }} \right) } d{x}^{\prime }$ and $B_{ j\pm 1/2}$ is the combination of derivatives of f at $x_{j\pm 1/2}$ by Taylor expansion. The second term in the previous formula can be further written as

$$\begin{aligned}&\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) q_{k,j\pm 1/2}^3} \\&\quad =\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) \left[ {H(x_{j\pm 1/2} )+A_k \Delta x^{3}+O\left( {\Delta x^{4}} \right) } \right] } \\&\quad =H(x_{j\pm 1/2} )\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) } +\Delta x^{3}\sum _{k=0}^3 {A_k \left( {\omega _{k,j\pm 1/2} -C_k } \right) } \\&\qquad +\sum _{k=0}^3 {\left( {\omega _{k,j\pm 1/2} -C_k } \right) O\left( {\Delta x^{4}} \right) }, \end{aligned}$$

where $A_{k}$ is the similar combination of derivatives of f at $x_{j\pm 1/2}$ by Taylor expansion for $q^{3}_{k}$. Thus,

$$\begin{aligned}&\frac{\hat{f}_{j+1/2} -\hat{f}_{j-1/2} }{\Delta x} \\&\quad =\frac{h(x_{j+1/2} )-h(x_{j-1/2} )}{\Delta x}+\left( {B_{j+1/2} -B_{j-1/2} } \right) O(\Delta x^{3}) \\&\qquad +\sum _{k=0}^3 {\frac{(\omega _{k,j+1/2} -C_k )q_{k,j+1/2} }{\Delta x}} -\sum _{k=0}^3 {\frac{(\omega _{k,j-1/2} -C_k )q_{k,j-1/2} }{\Delta x}} +O(\Delta x^{4}) \\&\quad =f_j^{\prime }+{\left( {H(x_{j+1/2} )\sum _{k=0}^3 {(\omega _{k,j+1/2} -C_k )} -H(x_{j-1/2} )\sum _{k=0}^3 {(\omega _{k,j-1/2} -C_k )} } \right) }/{\Delta x} \\&\qquad +\Delta x^{2}\sum _{k=0}^3 {A_k (\omega _{k,j+1/2} -\omega _{k,j-1/2} )} \\&\qquad +\left[ {\sum _{k=0}^3 {(\omega _{k,j+1/2} -C_k )} -\sum _{k=0}^3 {(\omega _{k,j-1/2} -C_k )} } \right] O(\Delta x^{3})+O(\Delta x^{4}) \end{aligned}$$

Hence the necessary and sufficient condition (NSC) for the fourth-order convergence is

$$\begin{aligned} \sum _{k=0}^3 {(\omega _{k,j\pm 1/2} -C_k )}= & {} O(\Delta x^{5}) \\ \sum _{k=0}^3 {A_k (\omega _{k,j+1/2} -\omega _{k,j-1/2} )}= & {} O(\Delta x^{2}) \\ \omega _{k,j\pm 1/2} -C_k= & {} O(\Delta x), \end{aligned}$$

which is similar to the fifth-order counterpart in Ref. [3].

Next we will show if the last one of NSC is satisfied, its second one will stand. Considering that $\omega _{k}$ is generally derived from ${ IS}_{k}$, which is comprised of f at dependent grid points {$x_{j-2}$, ..., $x_{j+3}$}, it is reasonable to assume,

$$\begin{aligned} \omega _{k,j+1/2} =C_k +g\left( {f_{j-2}, f_{j-1}, f_j, f_{j+1}, f_{j+2}, f_{j+3} } \right) \Delta x+O(\Delta x^{2}) \end{aligned}$$

where g stands for certain differentiable function with six independent variables as $g(y_{1}$, ...$y_{6})$. Then,

$$\begin{aligned} \omega _{k,j+1/2} -\omega _{k,j-1/2}= & {} \left[ {\sum _{i=1}^6 {\left( {\frac{\partial g}{\partial y_i }} \right) \Delta f_{j-4+i} } } \right] \Delta x+O(\Delta x^{2})\\= & {} \left[ {\sum _{i=1}^6 {\left( {\frac{\partial g}{\partial y_i }} \right) } } \right] \left( {\frac{\partial f}{\partial x}} \right) _j \Delta x^{2}+O(\Delta x^{2}), \end{aligned}$$

where $\Delta f_j =f_{j+1} -f_j$. So the second one of NSC is satisfied. Considering Eqs. (11) and (12), the last one of NSC is satisfied when ${ IS}_{k}$ is given by Eq. (22). Therefore the proposed indicator can make the nonlinear scheme achieve the fourth-order when critical points are not of concern.

Appendix 2

Here we will show that if a varying $p_{wr}$ is used as the exponent in Eq. (10), the accuracy relation will be preserved. First we notice

$$\begin{aligned} \left( {1+x} \right) ^{p_{wr} }=1+p_{wr} \cdot x+\cdots +\frac{p_{wr} \left( {p_{wr} -1} \right) \cdot \cdot \cdot (p_{wr} -n+1)}{n!}x^{n}+\cdots ,\hbox { for }\left| x \right| \le 1. \end{aligned}$$

Then from Eqs. (10) and (11), the following relation still stands as

$$\begin{aligned} \alpha _k =\frac{C_k^r }{D^{p_{wr} }}\left( {1+O\left( {\Delta x} \right) ^{R-r_{cs} }} \right) , \end{aligned}$$

where the meaning of symbols can be found in Sect. 3.1. Then,

$$\begin{aligned} \sum _l {\alpha _l } =\sum _l {\frac{C_l }{D^{p_{wr} }}\left( {1-O(\Delta x^{R-r_{cs} })} \right) } =\frac{1}{D^{p_{wr} }}-O(\Delta x^{R-r_{cs} }), \end{aligned}$$

and therefore

$$\begin{aligned} \omega _k= & {} \frac{\alpha _k }{\sum _l {\alpha _l } }=\frac{C_k^r \left( {1-O(\Delta x^{R-r_{cs} })} \right) }{1-O(\Delta x^{R-r_{cs} })}=C_k^r \left( {1-O(\Delta x^{R-r_{cs1}})} \right) \left( {1+O(\Delta x^{R-r_{cs} })+\cdots } \right) \\= & {} C_k^r +O(\Delta x^{R-r_{cs} }). \end{aligned}$$

Hence the accuracy requirement by Ref. [1] is satisfied, and $p_{wr}$ can be integrated into the weighted scheme used as the exponent in Eq. (10).

Appendix 3

When characteristic variables are used in the scheme, a diagonal matrix may need to be used for rescaling. This is because the threshold values of the algorithm in Sect. 3.4 are obtained when the following $L_{ref}$ is used to derive characteristic variables, so the thresholds might not work when other forms of characteristic variables are used. And this happens when different left eigenvector matrix is used like L below. In order to make the thresholds work for different characteristic variables, an operation is proposed by multiplying a diagonal matrix $\Lambda _{cv}$. The derivation of the matrix (Eq. 34) is explained as follows.

(1) First consider the following left eigenvector matrix $L_{ref}$ to compute characteristic variables, by using of which aforementioned threshold values work in computations.

$$\begin{aligned} L_{ref} =\left[ \begin{array}{ccccc} {\frac{1}{2a^{2}}\left( {\phi ^{2}+a\bar{\theta }} \right) } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) u+\bar{k}_x a} \right] } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) v+\bar{k}_y a} \right] } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) w+\bar{k}_z a} \right] } &{} {\frac{1}{2a^{2}}\left( {\gamma -1} \right) } \\ {\bar{k}_x \left( {1-\frac{\phi ^{2}}{a^{2}}} \right) +\bar{k}_z \frac{v}{a}-\bar{k}_y \frac{w}{a}} &{} {\frac{\bar{k}_x }{a^{2}}\left( {\gamma -1} \right) u} &{} {\frac{\bar{k}_x }{a^{2}}\left( {\gamma -1} \right) v-\frac{\bar{k}_z }{a}} &{} {\frac{\bar{k}_x }{a^{2}}\left( {\gamma -1} \right) w+\frac{\bar{k}_y }{a}} &{} {-\frac{\bar{k}_x }{a^{2}}\left( {\gamma -1} \right) } \\ {\bar{k}_y \left( {1-\frac{\phi ^{2}}{a^{2}}} \right) +\bar{k}_x \frac{w}{a}-\bar{k}_z \frac{u}{a}} &{} {\frac{\bar{k}_y }{a^{2}}\left( {\gamma -1} \right) u+\frac{\bar{k}_z }{a}} &{} {\frac{\bar{k}_y }{a^{2}}\left( {\gamma -1} \right) v} &{} {\frac{\bar{k}_y }{a^{2}}\left( {\gamma -1} \right) w-\frac{\bar{k}_x }{a}} &{} {-\frac{\bar{k}_y }{a^{2}}\left( {\gamma -1} \right) } \\ {\bar{k}_z \left( {1-\frac{\phi ^{2}}{a^{2}}} \right) +\bar{k}_y \frac{u}{a}-\bar{k}_x \frac{v}{a}} &{} {\frac{\bar{k}_z }{a^{2}}\left( {\gamma -1} \right) u-\frac{\bar{k}_y }{a}} &{} {\frac{\bar{k}_z }{a^{2}}\left( {\gamma -1} \right) v+\frac{\bar{k}_x }{a}} &{} {\frac{\bar{k}_z }{a^{2}}\left( {\gamma -1} \right) w} &{} {-\frac{\bar{k}_z }{a^{2}}\left( {\gamma -1} \right) } \\ {\frac{1}{2a^{2}}\left( {\phi ^{2}-a\bar{\theta }} \right) } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) u-\bar{k}_x a} \right] } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) v-\bar{k}_y a} \right] } &{} {-\frac{1}{2a^{2}}\left[ {\left( {\gamma -1} \right) w-\bar{k}_z a} \right] } &{} {\frac{1}{2a^{2}}\left( {\gamma -1} \right) } \\ \end{array} \right] \end{aligned}$$

where k represents $\xi$, $\eta$, or $\zeta$, a is the sound speed, $\varphi ^{2}=\frac{1}{2}\left( {\gamma -1} \right) \left( {u^{2}+v^{2}+w^{2}} \right)$, $\bar{\theta }=\frac{k_x u+k_y v+k_z w}{\sqrt{k_x^{2}+k_y^{2}+k_z^{2}}}$, $\bar{k}_x =\frac{k_x }{\sqrt{k_x^{2}+k_y^{2}+k_z^{2}}}$, and so it is with $\bar{k}_y$ and $\bar{k}_z$.

The inverse matrix of $L_{ref}$ is also presented for completeness:

$$\begin{aligned} L_{ref}^{-1} =\left[ {{\begin{array}{ccccc} 1&{} {\bar{k}_x }&{} {\bar{k}_y }&{} {\bar{k}_z }&{} 1 \\ {u-\bar{k}_x a}&{} {\bar{k}_x u}&{} {\bar{k}_y u+\bar{k}_z a}&{} {\bar{k}_z u-\bar{k}_y a}&{} {u+\bar{k}_x a} \\ {v-\bar{k}_y a}&{} {\bar{k}_x v-\bar{k}_z a}&{} {\bar{k}_y v}&{} {\bar{k}_z v+\bar{k}_x a}&{} {v+\bar{k}_y a} \\ {w-\bar{k}_z a}&{} {\bar{k}_x w+\bar{k}_y a}&{} {\bar{k}_y w-\bar{k}_x a}&{} {\bar{k}_z w}&{} {w+\bar{k}_z a} \\ {\frac{\varphi ^{2}+a^{2}}{\gamma -1}-\bar{\theta }a}&{} {\frac{\bar{k}_x \varphi ^{2}a}{\gamma -1}+\bar{k}_y wa-\bar{k}_z va}&{} {\frac{\bar{k}_y \varphi ^{2}a}{\gamma -1}+\bar{k}_z ua-\bar{k}_x wa}&{} {\frac{\bar{k}_z \varphi ^{2}a}{\gamma -1}+\bar{k}_x va-\bar{k}_y ua}&{} {\frac{\varphi ^{2}+a^{2}}{\gamma -1}+\bar{\theta }a} \\ \end{array} }} \right] \end{aligned}$$

(2) Next consider another matrix L to derive characteristic variables. The following relation stands: $L\left( \cdot \right) =LL_{ref}^{-1} L_{ref} \left( \cdot \right)$ or $L_{ref} \left( \cdot \right) =\left( {L_{ref}^{-1} L} \right) L\left( \cdot \right)$. Then the difference between the two versions of variables will be $LL_{ref}^{-1}$, i.e., $\Lambda _{cv}$. Take L used in the paper as an example, which is of the form:

$$\begin{aligned} L=\left[ \begin{array}{ccccc} {\frac{1}{\sqrt{2}a}\left( {\varphi ^{2}+a\bar{\theta }} \right) } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) u+\bar{k}_x a} \right] } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) v+\bar{k}_y a} \right] } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) w+\bar{k}_z a} \right] } &{} {\frac{1}{\sqrt{2}a}\left( {\gamma -1} \right) } \\ {\frac{\bar{k}_x }{a}\left( {a^{2}-\varphi ^{2}} \right) +\bar{k}_z v-\bar{k}_y w} &{} {\frac{\bar{k}_x }{a}\left( {\gamma -1} \right) u} &{} {\frac{\bar{k}_x }{a}\left( {\gamma -1} \right) v-\bar{k}_z } &{} {\frac{\bar{k}_x }{a}\left( {\gamma -1} \right) w+\bar{k}_y } &{} {-\frac{\bar{k}_x }{a}\left( {\gamma -1} \right) } \\ {\frac{\bar{k}_y }{a}\left( {a^{2}-\varphi ^{2}} \right) +\bar{k}_x w-\bar{k}_z u} &{} {\frac{\bar{k}_y }{a}\left( {\gamma -1} \right) u+\bar{k}_z } &{} {\frac{\bar{k}_y }{a}\left( {\gamma -1} \right) v} &{} {\frac{\bar{k}_y }{a}\left( {\gamma -1} \right) w-\bar{k}_x } &{} {-\frac{\bar{k}_y }{a}\left( {\gamma -1} \right) } \\ {\frac{\bar{k}_z }{a}\left( {a^{2}-\varphi ^{2}} \right) +\bar{k}_y u-\bar{k}_x v} &{} {\frac{\bar{k}_z }{a}\left( {\gamma -1} \right) u-\bar{k}_y } &{} {\frac{\bar{k}_z }{a}\left( {\gamma -1} \right) v+\bar{k}_x } &{} {\frac{\bar{k}_z }{a}\left( {\gamma -1} \right) w} &{} {-\frac{\bar{k}_z }{a}\left( {\gamma -1} \right) } \\ {\frac{1}{\sqrt{2}a}\left( {\varphi ^{2}-a\bar{\theta }} \right) } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) u-\bar{k}_x a} \right] } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) v-\bar{k}_y a} \right] } &{} {-\frac{1}{\sqrt{2}a}\left[ {\left( {\gamma -1} \right) w-\bar{k}_z a} \right] } &{} {\frac{1}{\sqrt{2}a}\left( {\gamma -1} \right) } \\ \end{array}\right] \end{aligned}$$

For completeness, the $L^{-1}$ is presented also as:

$$\begin{aligned} L^{-1}=\left[ \begin{array}{ccccc} {\frac{1}{\sqrt{2}a}} &{} {\frac{\bar{k}_x }{a}} &{} {\frac{\bar{k}_y }{a}} &{} {\frac{\bar{k}_z }{a}} &{} {\frac{1}{\sqrt{2}a}} \\ {\frac{1}{\sqrt{2}a}\left( {u-\bar{k}_x a} \right) } &{} {\frac{\bar{k}_x }{a}u} &{} {\frac{\bar{k}_y }{a}u+\bar{k}_z } &{} {\frac{\bar{k}_z }{a}u-\bar{k}_y } &{} {\frac{1}{\sqrt{2}a}\left( {u+\bar{k}_x a} \right) } \\ {\frac{1}{\sqrt{2}a}\left( {v-\bar{k}_y a} \right) } &{} {\frac{\bar{k}_x }{a}v-\bar{k}_z } &{} {\frac{\bar{k}_y }{a}v} &{} {\frac{\bar{k}_z }{a}v+\bar{k}_x } &{} {\frac{1}{\sqrt{2}a}\left( {v+\bar{k}_y a} \right) } \\ {\frac{1}{\sqrt{2}a}\left( {w-\bar{k}_z a} \right) } &{} {\frac{\bar{k}_x }{a}w+\bar{k}_y } &{} {\frac{\bar{k}_y }{a}w-\bar{k}_x } &{} {\frac{\bar{k}_z }{a}w} &{} {\frac{1}{\sqrt{2}a}\left( {w+\bar{k}_z a} \right) } \\ {\frac{1}{\sqrt{2}a}\left( {\frac{\varphi ^{2}+a^{2}}{\gamma -1}-\bar{\theta }a} \right) } &{} {\bar{k}_x \chi +\bar{k}_y w-\bar{k}_z v} &{} {\bar{k}_y \chi +\bar{k}_z u-\bar{k}_x w} &{} {\bar{k}_z \chi +\bar{k}_x v-\bar{k}_y u} &{} {\frac{1}{\sqrt{2}a}\left( {\frac{\varphi ^{2}+a^{2}}{\gamma -1}+\bar{\theta }a} \right) } \\ \end{array} \right] \end{aligned}$$

It can be found that: $L_{ref}^{-1} L={ diag}\left( {1\big /{\left( {\sqrt{2}a} \right) },1/a,\ldots ,1/a,1\big /{\left( {\sqrt{2}a} \right) }} \right)$, which turns to be $\Lambda _{cv}$ in Eq. (34).