Asymptotic Preserving Low Mach Number Accurate IMEX Finite Volume Schemes for the Isentropic Euler Equations

Abstract

In this paper, the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge–Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. The asymptotic accuracy is concerned with the closeness of the compressible solution with that of its incompressible counterpart in a low Mach number regime. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.

Appendix

First, let us introduce the following shorthands (see Fig. 7 for RK coefficients).

$$\begin{aligned} b^{(2)}_1&= {\tilde{\omega }}_2 {\tilde{c}}_{2} - \frac{1}{2}, \quad b^{(2)}_2= {\tilde{\omega }}_1 c_{1} + {\tilde{\omega }}_2 c_{2} + \omega _2 {\tilde{c}}_{2} - 1, \quad b^{(2)}_3=\omega _1c_{1}+\omega _{2} c_{2}-\frac{1}{2}, \nonumber \\ b^{(3)}_1&= \frac{1}{2} - \left\{ {\tilde{\omega }}_2 {\tilde{c}}_{2} \sum _{i=1}^{2}a_{i,i} + {\tilde{\omega }}_3 \sum _{j=1}^{3} ( {\tilde{a}}_{3,j} c_{j} + a_{3,j} {\tilde{c}}_{j}) + \omega _3 {\tilde{c}}_{2} {\tilde{a}}_{3,2} \right\} , \nonumber \\ b^{(3)}_2&= \frac{1}{6} - {\tilde{\omega }}_3 {\tilde{c}}_{2} {\tilde{a}}_{3,2} \nonumber \\ b^{(3)}_3&= \frac{1}{2} - \left\{ \sum _{i = 1}^{3 }{\tilde{\omega }}_i\sum _{j = 1}^{i } a_{i,j} c_{j} + \omega _2 {\tilde{c}}_{2}\sum _{i=1}^{2}a_{i,i}+ \omega _3\sum _{j = 1}^{3} ( {\tilde{a}}_{3,j} c_{j} + a_{3,j} {\tilde{c}}_{j}) \right\} , \nonumber \\ b^{(3)}_4&= \frac{1}{6} - \sum _{i,j=1}^{3} \omega _i a_{i,j} c_{j} \nonumber \\ b^{(4)}_1&= {\tilde{\omega }}_2 {\tilde{c}}_{2}\left( a_{1,1}\sum _{i=1}^2a_{i,i} + a_{2,2}^2\right) +{\tilde{\omega }}_3 \left( \sum _{j= 1}^2 {\tilde{a}}_{3,i} \sum _{i=1}^j c_{i} a_{j,i}+ a_{3,2} {\tilde{c}}_{2} \sum _{i=1}^2a_{i,i} \right. \nonumber \\&\quad \left. + a_{3,3} \sum _{i=1}^2 {\tilde{a}}_{3,i} c_{i} + \sum _{j=2}^3 a_{3,j} {\tilde{c}}_{j}\right) \nonumber \\&\quad + \omega _{3} {\tilde{a}}_{3,2} {\tilde{c}}_{2} \sum _{i=1}^3 a_{i,i} - \frac{1}{4} , \quad b^{(4)}_2 = {\tilde{\omega }}_{3} {\tilde{a}}_{3,2}{\tilde{a}}_{2,1} \sum _{i=1}^{3}a_{i,i} - \frac{1}{6} \nonumber \\ b^{(4)}_3&= \sum _{k=1}^3 {\tilde{\omega }}_{k} \sum _{j = 1}^k a_{k,j} \sum _{i = 1}^j a_{j,i} c_{i} + \sum _{k=2}^3\omega _{k} \sum _{j=1}^{k-1} {\tilde{a}}_{k,j} \sum _{i=1}^j a_{j,i} c_i + \sum _{k=2}^3 \omega _k {\tilde{a}}_{2,1} a_{k,2} \sum _{i=1}^2a_{i,i} \nonumber \\&+ \omega _{3} a_{3,3} \sum _{j=1}^2 ({\tilde{a}}_{3,j} c_j + a_{3,j+1} {\tilde{c}}_{j+1}) - \frac{1}{6}, \quad b^{(4)}_4= \sum _{k = 1}^3\omega _{k} \sum _{j=1}^k a_{k,j} \sum _{i = 1}^j a_{j,i} c_i - \frac{1}{24} \nonumber \\ \end{aligned}$$

(8.1)

Fig. 7

Double Butcher tableaux of IMEX-RK schemes. Top left: Euler (1,1,1), top middle: JIN(2,2,2), top right: PR(2,2,2), where $\delta _p=1-(1/2\gamma _p)$, bottom left: ARS(2,2,2), where $\gamma _a=1-(\sqrt{2}/2), \ \delta _a=1-(1/2\gamma _a)$, and bottom right: CN(2,2,2)

With the above notations, the entries in the matrices $\mathcal {B}^{(2)},B^{(3)}$ and $B^{(4)}$ read

$$\begin{aligned} \mathcal {B}_{1,1}^{(2)}&=\Delta t\left\{ b^{(2)}_1 ({{\underline{{\varvec{u}}}}} \cdot \nabla )^2+b^{(2)}_3\frac{{{\underline{a}}}^2}{\varepsilon ^2} \Delta \right\} +\frac{1}{2}\Delta x_k|{{{\underline{u}}}}_k|\partial _{x_k}^2, \quad \mathcal {B}_{1,2}^{(2)}=\Delta t{{\underline{\rho }}}b^{(2)}_2({{\underline{{\varvec{u}}}}}\cdot \nabla ) \nabla \cdot , \\ \mathcal {B}_{2,1}^{(2)}&=\Delta t\frac{{{\underline{a}}}^2}{{{\underline{\rho }}}\varepsilon ^2}b^{(2)}_2({{\underline{{\varvec{u}}}}} \cdot \nabla ) \nabla , \quad \mathcal {B}_{2,2}^{(2)}=\Delta t\left\{ b^{(2)}_1({{\underline{{\varvec{u}}}}} \cdot \nabla )^2+ b^{(2)}_3\frac{{{\underline{a}}}^2}{\varepsilon ^2}\nabla (\nabla \cdot ) \right\} +\frac{1}{2}\Delta x_k|{{{\underline{u}}}}_k|\partial _{x_k}^2\mathbb {I}_2. \\ B^{(3)}_{1,1}&=b^{(3)}_2 ({{\underline{{\varvec{u}}}}} \cdot \nabla )^3 + b^{(3)}_3\frac{{{\underline{a}}}^2}{\varepsilon ^2}({{\underline{{\varvec{u}}}}} \cdot \nabla ) \Delta , \quad B^{(3)}_{1,2}=b^{(3)}_1 {\underline{\rho }}({{\underline{{\varvec{u}}}}} \cdot \nabla )^2 \nabla \cdot + b^{(3)}_4 \frac{{{\underline{a}}}^2 {{\underline{\rho }}}}{\varepsilon ^2} \Delta \nabla \cdot , \\ B^{(3)}_{2,1}&=b^{(3)}_1 \frac{{{\underline{a}}}^2}{\varepsilon ^2} ({{\underline{{\varvec{u}}}}} \cdot \nabla )^2 \nabla + b^{(3)}_4 \frac{{{\underline{a}}}^4 }{{{\underline{\rho }}}\varepsilon ^4} \Delta \nabla , \quad B^{(3)}_{2,2}=b^{(3)}_2({{\underline{{\varvec{u}}}}} \cdot \nabla )^3 \mathbb {I}_2+b^{(3)}_3 \frac{{{\underline{a}}}^2}{\varepsilon ^2}({{\underline{{\varvec{u}}}}} \cdot \nabla ) \nabla \nabla \cdot . \\ B^{(4)}_{1,1}&= b^{(4)}_1 \frac{{{\underline{a}}}^2}{\varepsilon ^2}({{\underline{{\varvec{u}}}}} \cdot \nabla )^2 \Delta + b^{(4)}_4\frac{{{\underline{a}}}^4}{\varepsilon ^4} \Delta ^2 - \frac{1}{24} ({{\underline{{\varvec{u}}}}} \cdot \nabla )^4, \quad B^{(4)}_{1,2}=b^{(4)}_2{{\underline{\rho }}} ({{\underline{{\varvec{u}}}}} \cdot \nabla )^3 \nabla \cdot \\&\quad +b^{(4)}_3 \frac{{{\underline{a}}}^2 {{\underline{\rho }}}}{\varepsilon ^2} ({{\underline{{\varvec{u}}}}} \cdot \nabla )\Delta \nabla \cdot , \\ B^{(4)}_{2,1}&= b^{(4)}_2 \frac{{{\underline{a}}}^2}{{{\underline{\rho }}} \varepsilon ^2}({{\underline{{\varvec{u}}}}} \cdot \nabla )^3 \nabla + b^{(4)}_3\frac{{{\underline{a}}}^4 }{{{\underline{\rho }}}\varepsilon ^4}({{\underline{{\varvec{u}}}}} \cdot \nabla )\Delta \nabla , \ B^{(4)}_{2,2}=b^{(4)}_1 \frac{{{\underline{a}}}^2}{\varepsilon ^2}({{\underline{{\varvec{u}}}}} \cdot \nabla )^2 \nabla (\nabla \cdot )\\&\quad + b^{(4)}_4\frac{{{\underline{a}}}^4}{\varepsilon ^4} \nabla \Delta \nabla \cdot - \frac{1}{24} ({{\underline{{\varvec{u}}}}} \cdot \nabla )^4 \mathbb {I}_2. \end{aligned}$$