An Entropy Stable Finite Volume Scheme for the Equations of Shallow Water Magnetohydrodynamics

Abstract

In this work, we design an entropy stable, finite volume approximation for the shallow water magnetohydrodynamics (SWMHD) equations. The method is novel as we design an affordable analytical expression of the numerical interface flux function that exactly preserves the entropy, which is also the total energy for the SWMHD equations. To guarantee the discrete conservation of entropy requires a special treatment of a consistent source term for the SWMHD equations. With the goal of solving problems that may develop shocks, we determine a dissipation term to guarantee entropy stability for the numerical scheme. Numerical tests are performed to demonstrate the theoretical findings of entropy conservation and robustness.

Appendix: Flux Functions in Two Spatial Dimensions

We note that, for the discussion of the two dimension fluxes we suppress the i index on the multidimensional approximation for the purpose of clarity.

1.1 Entropy Conservative Flux

To develop the entropy conservative flux in the y-direction we first note that the entropy potential

$$\begin{aligned} \psi _y = \varvec{v}\cdot \varvec{g} - G = \frac{g}{2}h^2v_2 - hB_2(v_1B_1 + v_2B_2), \end{aligned}$$

(7.2)

where we have the entropy flux in the y-direction

$$\begin{aligned} G = gh^2v_2 + \frac{1}{2}\left( hv_1^2v_2 + hv_2^3 + hv_2B_1^2 - hv_2B_2^2\right) - hv_1B_1B_2. \end{aligned}$$

(7.3)

Note that the discrete entropy conservation condition (4.11) has the same structure in each Cartesian direction. Lastly, the source term contributes symmetrically to each direction. With a proof analogous of that for $\varvec{f}^{*,ec}$ in Sect. 5 we present the entropy conserving numerical flux for the y-direction.

Corollary 1

(Entropy Conserving Numerical Flux: y-direction) If we discretize the source term in the finite volume method to contribute to each element as

(7.4)

then we can determine a discrete, entropy conservative flux of the form

(7.5)

1.2 Entropy Stable Fluxes

Just as was done in Sect. 5.2 we can create 2D entropy stable flux functions. We, again, use the flux Jacobian altered by the Powell source term to create the dissipation term:

$$\begin{aligned} \widehat{\mathbf{B }} = \begin{bmatrix} 0&\quad 0&\quad 1&\quad 0&\quad 0 \\ -v_1v_2+B_1B_2&\quad v_2&\quad v_1&\quad -B_2&\quad 0 \\ gh - v_2^2 + B_2^2&\quad 0&\quad 2v_2&\quad 0&\quad -B_2 \\ v_1B_2 - v_2B_1&\quad -B_2&\quad B_1&\quad v_2&\quad 0 \\ 0&\quad 0&\quad 0&\quad 0&\quad v_2 \\ \end{bmatrix}, \end{aligned}$$

(7.6)

equipped with a full set of eigenvalues

$$\begin{aligned} \lambda _1 = v_2-c_g,\quad \lambda _2 = v_2-B_2,\quad \lambda _3 = v_2,\quad \lambda _4 = v_2+B_2,\quad \lambda _5 = v_2+c_g, \end{aligned}$$

(7.7)

and right eigenvectors

$$\begin{aligned} \widehat{\mathbf{R }}_y = \begin{bmatrix} 1&\quad 0&\quad 1&\quad 0&\quad 1 \\ v_1&\quad 1&\quad v_1&\quad 1&\quad v_1\\ v_2-c_g&\quad 0&\quad v_2&\quad 0&\quad v_2+c_g \\ B_1&\quad 1&\quad B_1&\quad -1&\quad B_1 \\ 0&\quad 0&\quad \frac{c_g^2}{B_2}&\quad 0&\quad 0 \end{bmatrix}, \quad \widehat{\mathbf{L }}_y = \widehat{\mathbf{R }}_y^{-1}, \end{aligned}$$

(7.8)

where $c_g^2 = gh+B_2^2$ is the magnetogravity wave speed.

Corollary 2

(Entropy Stable 1 (ES1): y-direction) If we apply the diagonal scaling matrix

$$\begin{aligned} \mathbf T _y = diag\left( \frac{c}{c_g\sqrt{2g}},\,\frac{c}{\sqrt{2g}},\,\frac{B_2}{c_g\sqrt{g}},\,\frac{c}{\sqrt{2g}},\,\frac{c}{c_g\sqrt{2g}}\right) , \end{aligned}$$

(7.9)

to the matrix of right eigenvectors $\widehat{\mathbf{R }}_y$ (7.8), then we obtain the Merriam identity [21] (Eq. 7.3.1 pg. 77)

$$\begin{aligned} \mathbf H = \widetilde{\mathbf{R }}_y\widetilde{\mathbf{R }}_y^T = \left( \widehat{\mathbf{R }}_y\mathbf T _y\right) \left( \widehat{\mathbf{R }}_y\mathbf T _y\right) ^T = \widehat{\mathbf{R }}_y\mathbf S _y\widehat{\mathbf{R }}_y^T, \end{aligned}$$

(7.10)

that relates the right eigenvectors of $\widehat{\mathbf{B }}$ to the entropy Jacobian matrix (4.4). For convenience, we introduce the diagonal scaling matrix $\mathbf S _y=\mathbf T \,^2\!\!\!\!_y$ in (7.10). We then have the guaranteed entropy stable flux interface contribution

$$\begin{aligned} \varvec{g}^{*,ES1} = \varvec{g}^{*,ec}-\frac{1}{2}\mathbf D \llbracket \varvec{u} \rrbracket =\varvec{g}^{*,ec} - \frac{1}{2} \widehat{\mathbf{R }}_y|\widehat{{\varvec{\varLambda }}}|\mathbf S _y\widehat{\mathbf{R }}_y^T\llbracket \varvec{q} \rrbracket . \end{aligned}$$

(7.11)

Remark 4

(Entropy Stable 2 (ES2): y-direction) If we choose the dissipation matrix to be

$$\begin{aligned} \mathbf D = |\lambda _{max}|\mathbf I , \end{aligned}$$

(7.12)

where $\lambda _{max}$ is the largest eigenvalue of the system from $\widehat{\mathbf{B }}$ and $\mathbf I$ is the identity matrix, then we obtain a local Lax-Friedrichs type interface stabilization

$$\begin{aligned} \begin{aligned} \varvec{g}^{*,ES2}&= \varvec{g}^{*,ec} - \frac{1}{2}|\lambda _{max}|\mathbf H \llbracket \varvec{q} \rrbracket . \end{aligned} \end{aligned}$$

(7.13)