Scale-Invariant Multi-resolution Alternative WENO Scheme for the Euler Equations

Abstract

The finite difference multi-resolution alternative weighted essentially non-oscillatory (MR-AWENO) scheme has been designed to solve hyperbolic conservation laws (Wang et al. in Comput Methods Appl Mech Eng 382:113853, 2021). However, the scheme is not scale-invariant and generates numerical oscillations near strong shocks. To overcome this issue, we design the scale-invariant Si-weights and the MR-AWENO-Si operator by including the new global smoothness indicator and the descaler. The resulting scale-invariant MR-AWENO-Si scheme captures discontinuity of any scale in the essentially non-oscillatory (ENO) way efficiently and robustly. We also demonstrate an interesting application of the scale-invariant scheme to achieve the well-balanced property for the compressible Euler equations under a gravitational potential field by modifying the numerical fluxes, reformulating the source terms, and enforcing the MR-AWENO-Si operator. A theoretical proof is given, and extensive one- and two-dimensional classical examples are used to verify the performance of the MR-AWENO-Si scheme in terms of accuracy, robustness, and well-balanced property.

Appendices

Appendix

Derivation of (66)

At the equilibrium state (57), one has the conservative variables ${{\textbf{Q}}}= (\rho ^e,0,{P^e}/{(\gamma -1)})^T$. The left and right eigenvectors of the Jacobian of the flux are

$$\begin{aligned} {\widetilde{{\textbf{L}}}}_{i+\frac{1}{2}} = \lambda _{i+\frac{1}{2}} \left( \begin{array}{rrr} 0 &{} -{\tilde{c}}_{i+\frac{1}{2}} &{} (\gamma -1) \\ 2\,{\tilde{c}}_{i+\frac{1}{2}}^2 &{} 0 &{} -2\,(\gamma -1) \\ 0 &{} {\tilde{c}}_{i+\frac{1}{2}} &{} (\gamma -1) \\ \end{array} \right) ,\quad {\widetilde{{\textbf{R}}}}_{i+\frac{1}{2}} = \left( \begin{array}{rrr} 1 &{} 1 &{} 1 \\[6pt] -{\tilde{c}}_{i+\frac{1}{2}} &{} 0 &{} {\tilde{c}}_{i+\frac{1}{2}} \\[6pt] {\widetilde{H}}_{i+\frac{1}{2}} &{} 0 &{} {\widetilde{H}}_{i+\frac{1}{2}} \\ \end{array} \right) , \end{aligned}$$

with $\lambda _{i+\frac{1}{2}} = \left( 2{\tilde{c}}^2_{i+\frac{1}{2}}\right) ^{-1}$.

Firstly, ${{\textbf{Q}}}_{i+\ell }$ are projected into the characteristic fields through the Roe-averaged left eigenvectors ${\widetilde{{\textbf{L}}}}_{i+\frac{1}{2}}$

$$\begin{aligned} {\widetilde{{{\textbf{Q}}}}}_{i+\ell }= {\widetilde{{\textbf{L}}}}_{i+\frac{1}{2}}{{\textbf{Q}}}_{i+\ell } = \left( \begin{array}{r} \lambda _{i+\frac{1}{2}}P^e_{i+\ell } \\ \rho ^e_{i+\ell } - 2\lambda _{i+\frac{1}{2}}P^e_{i+\ell } \\ \lambda _{i+\frac{1}{2}}P^e_{i+\ell } \end{array} \right) , \quad \ell =-2,\ldots ,3. \end{aligned}$$

Then the MR-AWENO interpolation procedure is used to compute the cell interface value ${\widetilde{{{\textbf{Q}}}}}_{i+\frac{1}{2}}^{-}$ as

$$\begin{aligned} \widetilde{{\textbf{Q}}}_{i+\frac{1}{2}}^{-} = {\mathcal {W}}_{i+\frac{1}{2}}^{-} [{\widetilde{{{\textbf{Q}}}}}] = \left( \begin{array}{c} {\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \quad \quad ~~~~\lambda _{i+\frac{1}{2}}P^e\right] \\ {\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \rho ^e - 2\lambda _{i+\frac{1}{2}}P^e\right] \\ {\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \quad \quad ~~~~\lambda _{i+\frac{1}{2}}P^e\right] \end{array}\right) , \end{aligned}$$

Finally, the interpolated values ${\widetilde{{{\textbf{Q}}}}}_{i+\frac{1}{2}}^{-}$ are projected back into the physical space via the right eigenvectors ${\widetilde{{\textbf{R}}}}_{i+\frac{1}{2}}$,

$$\begin{aligned} {{\textbf{Q}}}_{i+\frac{1}{2}}^{-} = {\widetilde{{\textbf{R}}}}_{i+\frac{1}{2}}\widetilde{{\textbf{Q}}}_{i+\frac{1}{2}}^{-} = \left( \begin{array}{c} 2{\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \,\lambda _{i+\frac{1}{2}}P^e\right] + {\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \rho ^e - 2\lambda _{i+\frac{1}{2}}P^e\right] \\ 0 \\ 2{\widetilde{H}}_{i+\frac{1}{2}}{\mathcal {W}}_{i+\frac{1}{2}}^{-}\left[ \,\lambda _{i+\frac{1}{2}}P^e\right] \end{array}\right) . \end{aligned}$$

The right cell interface ${{\textbf{Q}}}_{i+\frac{1}{2}}^{+}$ can be computed by a similar procedure.

Using the definition of enthalpy H (30) and $u=0$, one has ${\widetilde{H}}_{i+\frac{1}{2}} = {\tilde{c}}_{i+\frac{1}{2}}^2/(\gamma - 1)$ and the derivation of (66) is completed.