An Efficient Hybrid Method for Solving Euler Equations

Abstract

In this paper, a hybrid method suitable for solving the Euler equations using high order methods has been proposed. The method was implemented and validated with a seventh order WENO scheme in OpenFOAM^®. The hybrid method combines a simple MUSCL-type flux approach and a characteristic flux approach. In the MUSCL-type flux approach, the inviscid fluxes are computed using approximate Riemann solvers HLL and HLLC schemes based on the WENO-reconstructed state variables. Hence, this is dubbed as the VF (variable-based flux) approach. In critical regions where VF may produce spurious oscillations, a novel, low-dissipation HLL-based CF (characteristic flux) approach is applied. Critical regions were identified using a modified Bhagatwala–Lele shock sensor. The VF/CF hybrid method has been shown to produce high-resolution, essentially non-oscillatory results for a number of 1D and 2D problems at a fraction of the cost of a pure CF approach. Moreover, a 2D advection problem was designed to investigate the choice of state variables and flux schemes. The results have shed more light on the relation between Kelvin–Helmholtz roll-ups and numerical instabilities along slip lines.

Appendices

Appendix A

For each sub-stencil j, the fourth order polynomial approximation $\left( {\rho_{i + 1/2} } \right)_{j}$ and the three polynomial coefficients $a_{k,j}$ are determined from the cell averages in that sub-stencil using is a $4 \times 4$ reconstruction matrix ${\mathbf{R}}_{j}$ as shown below.

$$ \left( {\begin{array}{*{20}c} {\rho_{i + 1/2} } \\ {a_{1} } \\ {a_{2} } \\ {a_{3} } \\ \end{array} } \right)_{j} = {\mathbf{R}}_{j} \left( {\begin{array}{*{20}c} {\overline{{\rho_{i - 3 + j} }} } \\ {\overline{{\rho_{i - 2 + j} }} } \\ {\overline{{\rho_{i - 1 + j} }} } \\ {\overline{{\rho_{i + j} }} } \\ \end{array} } \right) $$

(A1.1)

For the case of uniform cells, the sub-stencil reconstruction matrices are given by:

$$ \begin{aligned} {\mathbf{R}}_{0} = \left[ {\begin{array}{*{20}c} { - \tfrac{1}{4}} & {\tfrac{13}{12}} & { - \tfrac{23}{12}} & {\tfrac{25}{12}} \\ { - \tfrac{11}{12}} & {\tfrac{15}{4}} & { - \tfrac{23}{4}} & {\tfrac{35}{12}} \\ { - \tfrac{3}{4}} & {\tfrac{11}{4}} & { - \tfrac{13}{4}} & {\tfrac{5}{4}} \\ { - \tfrac{1}{6}} & {\tfrac{1}{2}} & { - \tfrac{1}{2}} & {\tfrac{1}{6}} \\ \end{array} } \right] \\ {\mathbf{R}}_{1} = \left[ {\begin{array}{*{20}c} {\tfrac{1}{12}} & { - \tfrac{5}{12}} & {\tfrac{13}{12}} & {\tfrac{1}{4}} \\ {\tfrac{1}{12}} & { - \tfrac{1}{4}} & { - \tfrac{3}{4}} & {\tfrac{11}{12}} \\ { - \tfrac{1}{4}} & {\tfrac{5}{4}} & { - \tfrac{7}{4}} & {\tfrac{3}{4}} \\ { - \tfrac{1}{6}} & {\tfrac{1}{2}} & { - \tfrac{1}{2}} & {\tfrac{1}{6}} \\ \end{array} } \right] \\ {\mathbf{R}}_{2} = \left[ {\begin{array}{*{20}c} { - \tfrac{1}{12}} & {\tfrac{7}{12}} & {\tfrac{7}{12}} & { - \tfrac{1}{12}} \\ {\tfrac{1}{12}} & { - \tfrac{5}{4}} & {\tfrac{5}{4}} & { - \tfrac{1}{12}} \\ {\tfrac{1}{4}} & { - \tfrac{1}{4}} & { - \tfrac{1}{4}} & {\tfrac{1}{4}} \\ { - \tfrac{1}{6}} & {\tfrac{1}{2}} & { - \tfrac{1}{2}} & {\tfrac{1}{6}} \\ \end{array} } \right] \\ {\mathbf{R}}_{3} = \left[ {\begin{array}{*{20}c} {\tfrac{1}{4}} & {\tfrac{13}{12}} & { - \tfrac{5}{12}} & {\tfrac{1}{12}} \\ { - \tfrac{11}{12}} & {\tfrac{3}{4}} & {\tfrac{1}{4}} & { - \tfrac{1}{12}} \\ {\tfrac{3}{4}} & { - \tfrac{7}{4}} & {\tfrac{5}{4}} & { - \tfrac{1}{4}} \\ { - \tfrac{1}{6}} & {\tfrac{1}{2}} & { - \tfrac{1}{2}} & {\tfrac{1}{6}} \\ \end{array} } \right] \\ {\mathbf{R}}_{4} = \left[ {\begin{array}{*{20}c} {\tfrac{25}{12}} & { - \tfrac{23}{12}} & {\tfrac{13}{12}} & { - \tfrac{1}{4}} \\ { - \tfrac{35}{12}} & {\tfrac{23}{4}} & { - \tfrac{15}{4}} & {\tfrac{11}{12}} \\ {\tfrac{5}{4}} & { - \tfrac{13}{4}} & {\tfrac{11}{4}} & { - \tfrac{3}{4}} \\ { - \tfrac{1}{6}} & {\tfrac{1}{2}} & { - \tfrac{1}{2}} & {\tfrac{1}{6}} \\ \end{array} } \right] \\ \end{aligned} $$

(A1.2)

Appendix B

Given the third order polynomial

$$ \rho_{j} \left( x \right) = \left( {\rho_{i + 1/2} } \right)_{j} + \sum\limits_{k = 1}^{3} {a_{k,j} x^{k} } $$

(A2.1)

the smoothness indicators are defined as

$$ IS_{j,\;L} = \sum\limits_{k = 1}^{3} {\int\limits_{{x_{i} - \Delta x/2}}^{{x_{i} + \Delta x/2}} {\Delta x^{2k - 1} \left( {\frac{{d^{k} \rho_{j} }}{{dx^{k} }}} \right)^{2} dx} } \quad {\kern 1pt} IS_{j,\;R} = \sum\limits_{k = 1}^{3} {\int\limits_{{x_{i + 1} - \Delta x/2}}^{{x_{i + 1} + \Delta x/2}} {\Delta x^{2k - 1} \left( {\frac{{d^{k} \rho_{j} }}{{dx^{k} }}} \right)^{2} dx} } $$

(A2.2)

Since the reconstruction direction x is taken with respect to the face centre $\varvec{x}_{f}$ and non-dimensionalized with $\Delta x = \left\| {\varvec{x}_{\text{P}} - \varvec{x}_{\text{N}} } \right\|$, the above expressions can be simplified and generalized as follows.

$$ IS_{j,\;K} = r\sum\limits_{k = 1}^{3} {\int\limits_{0}^{r} {\left( {\frac{{d^{k} \rho_{j} }}{{dx^{k} }}} \right)^{2} dx} } \quad {\text{where}}\quad r = \left\{ {\begin{array}{*{20}c} { - 1,} & {{\text{if }}K = L} \\ { + 1,} & {{\text{f }}K = R} \\ \end{array} } \right. $$

(A2.3)

Substituting Eq. (A2.1) into Eq. (A2.3), the expression in Eq. (13) can be obtained as follows.

$$ \begin{aligned} IS_{j,\;K} &= r\left[ {\int\limits_{0}^{r} {\left( {a_{0,\;j} + 2a_{1,\;j} x + 3a_{2,\;j} x^{2} } \right)^{2} dx} + \int\limits_{0}^{r} {\left( {2a_{1,\;j} + 6a_{2,\;j} x} \right)^{2} dx} + \int\limits_{0}^{r} {\left( {6a_{2,\;j} } \right)^{2} dx} } \right] \\ & = r\left\{ \begin{array}{l} \left[ {a_{0,\;j}^{2} x + 2a_{0,\;j} a_{1,\;j} x^{2} + 2a_{0,\;j} a_{2,\;j} x^{3} + \tfrac{4}{3}a_{1,\;j}^{2} x^{3} + 3a_{1,\;j} a_{2,\;j} x^{4} + \tfrac{9}{5}a_{2,\;j}^{2} x^{5} } \right]_{0}^{r} \hfill \\ + \left[ {4a_{1,\;j}^{2} x + 12a_{1,\;j} a_{2,\;j} x^{2} + 12a_{2,\;j}^{2} x^{3} } \right]_{0}^{r} + \left[ {36a_{2,\;j}^{2} x} \right]_{0}^{r} \hfill \\ \end{array} \right\} \\ & = a_{0,\;j}^{2} + 2ra_{0,\;j} a_{1,\;j} + 2a_{0,\;j} a_{2,\;j} + \tfrac{16}{3}a_{1,\;j}^{2} + 15ra_{1,\;j} a_{2,\;j} + \tfrac{249}{5}a_{2,\;j}^{2} \\ &= \left( {a_{0,\;j} + ra_{1,\;j} + a_{2,\;j} } \right)^{2} + \tfrac{13}{3}\left( {a_{1,\;j} + \tfrac{3}{2}ra_{2,\;j} } \right)^{2} + \tfrac{781}{20}a_{2,\;j}^{2} \\ \end{aligned} $$

(A2.4)

Note that in the third and fourth lines of the above derivation, the following identity was used to simplify the result.

$$ r^{n} = \left\{ {\begin{array}{*{20}c} {1,} & {{\text{if }}n{\text{ is even}}} \\ {r,} & {{\text{if }}n{\text{ is odd}}} \\ \end{array} } \right. $$