A Characteristic-wise Alternative WENO-Z Finite Difference Scheme for Solving the Compressible Multicomponent Non-reactive Flows in the Overestimated Quasi-conservative Form

Abstract

The fifth, seventh and ninth order characteristic-wise alternative weighted essentially non-oscillatory (AWENO) finite difference schemes are applied to the fully conservative (FC) form and the overestimated quasi-conservative (OQC) form of the compressible multicomponent flows. Several linear and nonlinear numerical operators such as the linear Lax–Friedrichs operator and linearized nonlinear WENO operator and their mathematical properties are defined in order to build a general mathematical (numerical) framework for identifying the necessary and sufficient conditions required in maintaining the equilibriums of certain physical relevant properties discretely. In the case of OQC form, the AWENO scheme with the modified flux can be rigorously proved to maintain the equilibriums of velocity, pressure and temperature. Furthermore, we also show that the FC form cannot maintain the equilibriums without an additional advection equation of auxiliary variable involving the specific heat ratio. Extensive one- and two-dimensional classical benchmark problems, such as the moving material interface problem, multifluid shock-density interaction problem and shock-R22-bubble interaction problem, verify the theoretical results and also show that the AWENO schemes demonstrate less dissipation error and higher resolution than the classical WENO-Z scheme in the splitting form (Nonomura and Fujii in J Comput Phys 340:358–388, 2017).

Appendices

Appendix

A  Eigensystem of Fully Conservative Form

For the fully conservative form, the left and right eigenvectors of the Jacobian $\mathbf {A}=\frac{\partial \mathbf {F}}{\partial \mathbf {Q}}$ of the flux $\mathbf {F(\mathbf {Q})}$ are

$$\begin{aligned} \begin{aligned} \mathbf {L}=&\left[ \begin{array}{c} \mathbf {l_{1}}\\ \mathbf {l_{2}}\\ \mathbf {l_{3}}\\ \mathbf {l_{4}} \end{array} \right] = \left[ \begin{array}{cccc} \frac{1}{2}\left( b_{2}+b_{4}+\frac{u}{c}\right) &{} \quad -\frac{1}{2}\left( b_{1}u+\frac{1}{c}\right) &{} \quad \frac{1}{2}b_{1} &{} \quad -\frac{1}{2}b_{3}\\ 1-b_{2}-b_{4} &{} \quad b_{1}u &{} \quad -b_{1} &{} \quad b_{3} \\ \frac{1}{2}\left( b_{2}+b_{4}-\frac{u}{c}\right) &{} \quad -\frac{1}{2}\left( b_{1}u-\frac{1}{c}\right) &{} \quad \frac{1}{2}b_{1} &{} \quad -\frac{1}{2}b_{3}\\ -Y_{1} &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{array} \right] ,\\ \mathbf {R}=&\left[ \mathbf {r_{1}},~ \mathbf {r_{2}},~ \mathbf {r_{3}},~ \mathbf {r_{4}} \right] = \left[ \begin{array}{cccc} 1 &{} \quad 1 &{} \quad 1 &{} \quad 0\\ u-c &{} \quad u &{} \quad u+c &{} \quad 0\\ H-uc &{} \quad \frac{1}{2}u^{2} &{} \quad H+uc &{} \quad \frac{1}{\rho }\frac{\partial e}{\partial Y_{1}}\\ Y_{1} &{} \quad Y_{1} &{} \quad Y_{1} &{} \quad 1 \end{array} \right] , \end{aligned} \end{aligned}$$

(48)

and their corresponding eigenvalues are ${{\varvec{\Lambda }}}=\left[ \begin{array}{cccc} \lambda _{1},&\lambda _{2},&\lambda _{3},&\lambda _{4} \end{array} \right] = \left[ \begin{array}{cccc} u-c,&u,&u+c,&u \end{array} \right] ,$ where the sound speed c, the enthalpy H, $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$ and $\frac{\partial e}{\partial Y_{1}}$ are

$$\begin{aligned} c= & {} \sqrt{\frac{\gamma P}{\rho }},~H=\Gamma c^2+\frac{u^2}{2},~ b_{1}= \frac{1}{\Gamma c^{2}},~b_{2}= \frac{u^{2}b_{1}}{2},~b_{3}= b_{1}\frac{1}{\rho }\frac{\partial e}{\partial Y_{1}},~b_{4}= b_{3}Y_{1}, \end{aligned}$$

(49)

$$\begin{aligned} \frac{\partial e}{\partial Y_{1}}= & {} PM\left( \frac{\Gamma _1 - \Gamma }{M_1} - \frac{\Gamma _0-\Gamma }{M_0}\right) ~~~ \text {with}~~~\Gamma _m = \frac{1}{\gamma _m - 1},~m = 0,1. \end{aligned}$$

(50)

B  Eigensystem of Overestimated Quasi-Conservative Form

In the overestimated quasi-conservative form, it is impossible to define the flux Jacobian since the system (7) is no longer expressed in a conservative form and hence, the corresponding eigensystem cannot be derived easily as in the FC form. Therefore, the left and right eigenvectors are evaluated via the modified flux Jacobian $\mathbf {B}=\mathbf {M}\frac{\partial \mathbf {F}}{\partial \mathbf {Q}}$ and they are

$$\begin{aligned} \begin{aligned} \mathbf {L}=&\left[ \begin{array}{c} \mathbf {l_{1}}\\ \mathbf {l_{2}}\\ \mathbf {l_{3}}\\ \mathbf {l_{4}}\\ \mathbf {l_{5}}\\ \end{array} \right] = \left[ \begin{array}{ccccc} \frac{1}{2}\left( b_{2}+\frac{u}{c}\right) &{} \quad -\frac{1}{2}\left( b_{1}u+\frac{1}{c}\right) &{} \frac{1}{2}b_{1} &{} \quad 0&{} \quad -\frac{1}{2}b_{1}P\\ 1-b_{2}&{} \quad b_{1}u &{} \quad -b_{1} &{} \quad 0 &{} \quad b_{1}P \\ \frac{1}{2}\left( b_{2}-\frac{u}{c}\right) &{} \quad -\frac{1}{2}\left( b_{1}u-\frac{1}{c}\right) &{} \frac{1}{2}b_{1} &{}0 \quad &{} -\frac{1}{2}b_{1}P\\ -Y_{1} &{} \quad 0 &{} \quad 0 &{}1 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \end{array} \right] ,\\ \mathbf {R}=&\left[ \mathbf {r_{1}},~ \mathbf {r_{2}},~ \mathbf {r_{3}},~ \mathbf {r_{4}},~ \mathbf {r_{5}} \right] = \left[ \begin{array}{ccccc} 1 &{} \quad 1 &{} \quad 1 &{} \quad 0 &{} \quad 0\\ u-c &{} \quad u &{} \quad u+c &{} \quad 0 &{} \quad 0\\ H-uc &{} \quad \frac{1}{2}u^{2} &{} \quad H+uc &{} \quad 0 &{} \quad P\\ Y_{1} &{} \quad Y_{1} &{} \quad Y_{1} &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 1 \\ \end{array} \right] , \end{aligned} \end{aligned}$$

(51)

and their corresponding eigenvalues are ${\varvec{\Lambda }}= {[}\lambda _1,~\lambda _2,~\lambda _3,~\lambda _4,~\lambda _5{]} = {[}u-c,~u,~u+c,~u,~u{]}$.

C  The Coefficients $d_{n}^{2k}$ for the Different Orders of Approximation in the AWENO Scheme

Table 3 The values of $d_{n}^{2k}\times c_{2k}$ for the fifth order $(r=3)$, seventh order $(r=4)$ and ninth order $(r=5)$ AWENO schemes.

D  The Derivations of (23) and (25)

In this section, we give the derivations of (23) and (25) in a general case. After that, the results of (23) and (25) in the specific case for constant u and P are given.

We denote a vector $\tilde{\mathbf {Q}}_{j+l}$, for any fixed grid point j,

$$\begin{aligned} \tilde{\mathbf {Q}}_{j+l}=\left[ q_1,~ q_2,~q_3,~q_4,~q_5\right] ^{T}_{j+l},~l=-(r-1),\ldots ,r, \end{aligned}$$

(52)

where $q_i=\tilde{\mathbf {l}}_{i}\mathbf {Q}_{j+l},~i=1,\ldots ,5$ are the characteristic variables.

By substituting $\tilde{b}_{2}= \frac{\tilde{u}^{2}\tilde{b}_{1}}{2}$ and the equation of state, $\rho e =\Gamma P+\frac{1}{2}\rho u^{2}$ into $q_i=\tilde{\mathbf {l}}_{i}\mathbf {Q}_{j+l}$, and by defining the operator $\delta f=f-\tilde{f}$, one has

$$\begin{aligned} \left[ \begin{array}{c} q_1\\ q_2\\ q_3\\ q_4\\ q_5 \end{array}\right] =\left[ \begin{array}{c} -\frac{1}{2}\frac{\rho }{\tilde{c}}\delta u+\frac{1}{4}\tilde{b}_{1}\rho (\delta u)^2+\frac{1}{2}\tilde{b}_{1}\Gamma \delta P\\ \rho -\frac{1}{2}\tilde{b}_{1}\rho (\delta u)^2-\tilde{b}_{1}\Gamma \delta P\\ \frac{1}{2}\frac{\rho }{\tilde{c}}\delta u+\frac{1}{4}\tilde{b}_{1}\rho (\delta u)^2+\frac{1}{2}\tilde{b}_{1}\Gamma \delta P\\ \rho \delta Y_1\\ \Gamma \end{array}\right] , \end{aligned}$$

(53)

where $\tilde{b}_{1}$ and $\tilde{c}$ can be found in (49).

The WENO reconstructed conservative variables at the cell boundary $\mathbf {Q}^{\pm }_{j+\frac{1}{2}}$ are calculated as follows

$$\begin{aligned} \mathbf {Q}^{\pm }_{j+\frac{1}{2}}=\tilde{\mathbf {R}}\tilde{\mathbf {Q}}^{\pm }_{j+\frac{1}{2}} = \left[ \begin{array}{c} q_1+q_2+q_3\\ (\tilde{u}-\tilde{c})q_1+\tilde{u}q_2+(\tilde{u}+\tilde{c})q_3 \\ (\tilde{H}-\tilde{u}\tilde{c})q_1+\frac{1}{2}\tilde{u}^{2}q_2+(\tilde{H}+\tilde{u}\tilde{c})q_3+\tilde{P}q_5\\ \tilde{Y}_{1}q_1+\tilde{Y}_{1}q_2+\tilde{Y}_{1}q_3+q_4\\ q_5 \end{array} \right] ^{\pm }_{j+\frac{1}{2}}. \end{aligned}$$

(54)

1.1 D.1  Special Case for Constant Velocity u and Pressure P

In the case of constant u and P, one has $\delta u=\delta P=0$, and

$$\begin{aligned} \begin{aligned} \tilde{\mathbf {Q}}_{j+l}&~=\left[ 0,~ \rho ,~0,~\rho \delta Y_1,~\Gamma \right] ^{T}_{j+l},~l=-(r-1),\ldots ,r,\\ \tilde{\mathbf {Q}}^{\pm }_{j+\frac{1}{2}}&~=\left[ \begin{array}{c} q_1\\ q_2\\ q_3\\ q_4\\ q_5 \end{array} \right] ^{\pm }_{j+\frac{1}{2}}=W^{\pm }_{j+\frac{1}{2}}\left[ \begin{array}{c} q_1\\ q_2\\ q_3\\ q_4\\ q_5 \end{array}\right] =\left[ \begin{array}{l} 0\\ W^{\pm }_{j+\frac{1}{2}}\left[ \rho \right] \\ 0\\ W^{\pm }_{j+\frac{1}{2}}\left[ \rho \delta Y_1\right] \\ W^{\pm }_{j+\frac{1}{2}}\left[ \Gamma \right] \end{array}\right] ,\\ \mathbf {Q}^{\pm }_{j+\frac{1}{2}}&=\left[ \begin{array}{l} q_{2}\\ uq_{2}\\ \frac{1}{2}u^{2}q_{2}+Pq_{5}\\ \tilde{Y}_{1}q_{2}+q_{4}\\ q_{5} \end{array} \right] _{j+\frac{1}{2}}^{\pm }. \end{aligned} \end{aligned}$$

(55)