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HLLD Algorithm

Overview

This is a step through of the HLLD Riemann solver I’m going to add to my hydro-sandbox code. The specific algorithm is from Miyoshi & Kusano 2005 which is the original paper where the HLLD Riemann solver was introduced. This will serve as practice and figuring out any kinks before I add the HLLD solver (along with VL+CT) to Cholla.

Overall the algorithm is very similar to the HLLC algorithm detailed in my earlier post on the topic. However a lot of the details have changed; for example instead of the sound speed \( c \) we must now compute the speed of either the fast or slow magnetosonic wave \( c_{f,s} \). Similar small changes are common so be careful not to miss them. Note that the HLLD Riemann solver does not include all 7 MHD waves; It ignores the slow magnetosonic wave.

Glossary of Symbols

Note: since magnetic fields are inherently 3D due to the use of the outproduct even the 1D algorithm I write out here uses all three dimensions. The first dimension is always in the \( x \) direction however, again because of the cross product, the active elements of the magnetic field are in the \( y \) and \( z \) directions.

  • \(k \), Subscript to indicate left or right. i.e. \( k = L \ \text{or}\ R \)
  • \(j \), Subscript to indicate which dimension we’re using. i.e. \( j = x, y, \text{or}\; z \)
  • \(* \), Superscript that indicates if the variable is the value within the star region, ie the region between the fast magnetosonic waves and the Alfvén waves.
  • \(** \), Superscript that indicates if the variable is the value within the star region, ie the region between the Alfvén waves and the contact discontinuity.
  • \( \vec{F} \), Flux
  • \(\rho \), Density
  • \(p \), Pressure
  • \(E \), Energy
  • \(v \), Velocity
  • \(B \), Magnetic Field
  • \(p_T \), Total pressure
  • \(S_L \), Approximation of the speed of the left moving magnetosonic wave
  • \(S_L^* \), Approximation of the speed of the left moving Alfvên wave
  • \(S_M \), Approximation of the speed of the center (contact) wave speed
  • \(S_R \), Approximation of the speed of the right moving magnetosonic wave
  • \(S_R^* \), Approximation of the speed of the right moving Alfvên wave
  • \(c \), Sound speed
  • \(\vec{U} \), Vector of conserved variables

Glossary of New Equations

MHD comes along with many modifications to the various basic equations that we’re used to; new terms in the pressure and energy equations, new wave speeds, new eigenvalues etc. I summarize the ones I plan to use here.

Conservative Variable Vector and Flux

\[\vec{U}_k = \begin{bmatrix} \rho_k \\ \rho_k v_{x,k} \\ \rho_k v_{y,k} \\ \rho_k v_{z,k} \\ B_x \\ B_y \\ B_z \\ E \end{bmatrix} \text{,}\; \; \vec{F}_k = \begin{bmatrix} \rho_k v_{x,k} \\ \rho_k v_{x,k}^2 + p_{T,k} - B_{x,k}^2 \\ \rho v_{x,k} v_{y,k} - B_{x,k} B_{y,k} \\ \rho v_{x,k} v_{z,k} - B_{x,k} B_{z,k} \\ 0 \\ B_{y,k} v_{x,k} - B_{x,k} v_{y,k} \\ B_{z,k} v_{x,k} - B_{x,k} v_{z,k} \\ v_{x,k} \left( E_k + p_{T,k} \right) - B_{x,k} \left( \vec{v_k} \cdot \vec{B_k} \right) \end{bmatrix}\]

Eigenvalues

\[\lambda_{2,6} = v_x \mp c_a, \; \lambda_{1,7} = v_x \mp c_f, \; \lambda_{3,5} = v_x \mp c_s, \; \lambda_{4} = v_x\]

Alfvên Wave Speed

\[c_a = \frac{\mid B_x \mid}{\sqrt{\rho}}\]

Fast & Slow Magnetosonic Wavespeeds

\[c_{f,s} = \sqrt{\frac {\gamma p + \mid \vec{B} \mid^2 \pm \sqrt{\left( \gamma p \;+ \mid \vec{B} \mid^2 \right)^2 - 4\gamma p B_x^2 } } {2\rho}}\]

Pressure

\[p = \left( \gamma - 1 \right) \left( E - \frac{1}{2} \rho \mid \vec{v}\mid ^2 - \frac{1}{2} \mid \vec{B} \mid ^2 \right)\] \[p_T = p_{Total} = p + \frac{1}{2} \mid \vec{B} \mid ^2\]

Energy

\[E = \frac{p}{\gamma - 1} + \frac{1}{2}\left( \rho \mid \vec{v} \mid ^2 + \mid \vec{B} \mid ^2\right) = \frac{p_T - \frac{1}{2}\mid \vec{B} \mid^2 }{\gamma - 1} + \frac{1}{2}\left( \rho \mid \vec{v} \mid ^2 + \mid \vec{B} \mid ^2\right)\]

Which Variables are the Same Between States

\[S_M = v_{x,L}^* = v_{x,L}^{**} = v_{x,R}^* = v_{x,R}^{**}\] \[p_T^* = p_{T_L}^* = p_{T_L}^{**} = p_{T_R}^* = p_{T_R}^{**}\] \[\rho_k^* = \rho_k^{**}\] \[\vec{v}_R^{**} = \vec{v}_L^{**} = \vec{v}^{**} \text{if}\; B_x = 0\] \[\vec{B}_R^{**} = \vec{B}_L^{**} = \vec{B}^{**}\; \text{if}\; B_x = 0\]

Algorithm

Our goal is to choose the correct HLLD flux depending on the interface states.

1. Compute Acoustic & Contact Wave Speeds

First we need to compute \( S_L, S_L^*, S_M, S_R^*, \text{and} S_R \) using the below equations.

Computing \( S_L \) and \( S_R \)

We can use either of the below approximations for \( S_L \text{and} S_R \) (there are other options as well, these were just the two given in Miyoshi & Kusano 2005)

\[S_L = \min(\lambda_{min}(\vec{U}_L), \lambda_{min}(\vec{U}_R))\] \[S_R = \max(\lambda_{max}(\vec{U}_L), \lambda_{max}(\vec{U}_R))\]

Where \( \lambda_{min} \) is the smallest eigenvalue and \( \lambda_{max} \) is the largest.

OR

\[S_L = \min(v_{x,L}, v_{x,R}) - \max(c_{f,L}, c_{f,R})\] \[S_R = \max(v_{x,L}, v_{x,R}) + \max(c_{f,L}, c_{f,R})\]

Computing \( S_L^* \) and \( S_R^* \)

Note that the second term is the Alfvên speed in the star state.

\[S_L^* = S_M - \frac{\mid B_x \mid}{\sqrt{\rho_L^*}}\] \[S_R^* = S_M + \frac{\mid B_x \mid}{\sqrt{\rho_R^*}}\]

where

\[\rho_k^* = \rho_k \frac{S_k - v_{x,k}}{S_k - S_M}\]

Computing \( S_M \)

\[S_M = \frac {\rho_R v_{x,R} \left( S_R - v_{x,R}\right) - \rho_L v_{x,L} \left( S_L - v_{x,L}\right) + p_{T_L} - p_{T_R}} {\rho_R \left( S_R - v_{x,R}\right) - \rho_L \left( S_L - v_{x,L}\right)}\]

2. Determine Which State We’re In

Use the equation below with the wave speeds to determine which state we’re in. If we’re in a non-star state go to step 3a, for star states go to step 3b, and for double star states go to step 3c.

\[\vec{F}_{HLLC} = \begin{cases} \vec{F}_L & \text{if}\ 0 < S_L \\ \\ \vec{F}_L^* & \text{if}\ S_L \leq 0 < S_L^* \\ \\ \vec{F}_L^{**} & \text{if}\ S_L^* \leq 0 < S_M \\ \\ \vec{F}_R^{**} & \text{if}\ S_M \leq 0 < S_R^* \\ \\ \vec{F}_R^* & \text{if}\ S_R^* \leq 0 \leq S_R \\ \\ \vec{F}_R & \text{if}\ S_R < 0 \end{cases}\]

3. Compute & Return the Fluxes

3a. \( F_L \) or \( F_R \) State

This is the region outside of either magnetosonic wave. The flux is easy, simply compute fluxes with the below equation and return.

\[\vec{F}_k = \begin{bmatrix} \rho_k v_{x,k} \\ \rho_k v_{x,k}^2 + p_{T,k} - B_{x,k}^2 \\ \rho v_{x,k} v_{y,k} - B_{x,k} B_{y,k} \\ \rho v_{x,k} v_{z,k} - B_{x,k} B_{z,k} \\ 0 \\ B_{y,k} v_{x,k} - B_{x,k} v_{y,k} \\ B_{z,k} v_{x,k} - B_{x,k} v_{z,k} \\ v_{x,k} \left( E_k + p_{T,k} \right) - B_{x,k} \left( \vec{v_k} \cdot \vec{B_k} \right) \end{bmatrix}\]

3b. \( F_L^\) or \(F_R^ \) State

This is the region between the fast magnetosonic waves and the Alfvên waves. These fluxes are more complicated and require the use of the fluxes from step 3a as well. The algorithm is exactly the same for the left and right star states so the subscript \( k \) is used to indicate \( R \) or \( L \).

\[\vec{F}_k^* = \vec{F}_k + S_k \left( \vec{U}_k^* - \vec{U}_k \right)\]

and we find the components of \(\vec{U}_k^* \) with the following equations. \( \rho^*_k \) was already computed when finding \( S_k^* \) and \( v_{x,k}^* = S_M \).

Pressure is a bit more complicated. Miyoshi & Kusano 2005 show that, under their assumptions, \( p^{*}_{T,L} = p^{**}_{T,L} = p^{*}_{T,R} = p^{**}_{T,R} = p^{*}_{T} = p^{**}_{T}\)

\[p_T^* = p_{T,L} + \rho_L \left( S_L - v_{x,L} \right) \left( S_M - v_{x,L} \right) = p_{T,R} + \rho_R \left( S_R - v_{x,R} \right) \left( S_M - v_{x,R} \right)\]

They also note that

\[p_T^* = \frac { \rho_R p_{T_L} \left( S_R - v_{x,R} \right) - \rho_L p_{T_R} \left( S_L - v_{x,L} \right) + \rho_L \rho_R \left( S_R - v_{x,R} \right) \left( S_L - v_{x,L} \right) \left( v_{x,R} - v_{x,L} \right) } {\rho_R \left( S_R - v_{x,R} \right) - \rho_L \left( S_L - v_{x,L} \right)}.\]

And so

\[E_k^* = \frac { E_k \left( S_k - v_{x,k} \right) - p_{T,k} v_{x,k} + p_T^* S_M + B_x \left( \vec{v}_k \cdot \vec{B}_k - \vec{v}_k^* \cdot \vec{B}_k^* \right) } {S_k - S_M}.\]

We can find the \( y \) and \( z \) components of \( \vec{v} \) and \(\vec{B} \) with the follwing equations. Note these equations can give a term that is \( \frac{0}{0} \) if \( S_M = v_{x,k}\), \(S_k = v_{x,k} \pm c_{f,k}\), \(B_{y,k} = B_{z,k} = 0\), and \(B_x^2 \ge \gamma p_k \). If this is the case then there is no shock across \( S_k \), so set \( \vec{v}_k^* = \vec{v}_k\), \(B_{y,k} = B_{z,k} = 0\), \(\rho_k^* = \rho_k\), and \( p_{T,k}^* = p_{T,k} \).

\[v_{j,k}^* = v_{j,k} - B_x B_{j,k} \frac {S_M - v_{x,k}} {\rho_k \left( S_k - v_{x,k} \right) \left( S_k - S_M \right) - B_x^2 }\] \[B_{j,k}^* = B_{j,k} \frac {\rho_k \left( S_k - v_{x,k} \right)^2 - B_x^2 } {\rho_k \left( S_k - v_{x,k} \right) \left( S_k - S_M \right) - B_x^2 }\]

Note that the denominators are the same

3c. \( F_L^{**} \) or \( F_R^{**} \) State

This is the region between the Alfvên waves and the contact discontinuity (entropy wave). These fluxes require the fluxes from step 3a and 3b. The algorithm is exactly the same for the left and right star states so the subscript \( k \) is used to indicate \( R \) or \( L \).

\[\vec{F}_k^{**} = \vec{F}_k^* + S_k^{*} \left( \vec{U}_k^{**} - \vec{U}_k^* \right)\]

We already know some of the state variables from previous computations

\[v_{x,k}^{**} = S_M\] \[p_T^{**} = p_T^{*}\] \[\rho_k^{**} = \rho_k^*\]

So all we need to compute directly is \( \; v_y^{**} \), \( \; v_z^{**} \), \( \; B_y^{**} \), \( \; B_z^{**} \), and \( \; E_k^{**} \)

\[v_j^{**} = \frac { v_{j,L}^* \sqrt{\rho_L^*} + v_{j,R}^* \sqrt{\rho_R^*} + \left( B_{j,R}^* - B_{j,L}^* \right) \text{sign}\left( B_x \right) } {\sqrt{\rho_L^*} + \sqrt{\rho_R^*}}\] \[B_j^{**} = \frac { B_{j,R}^* \sqrt{\rho_L^*} + B_{j,L}^* \sqrt{\rho_R^*} + \left( v_{j,R}^* - v_{j,L}^* \right) \sqrt{\rho_L^* \rho_R^*} \text{sign}\left( B_x \right) } {\sqrt{\rho_L^*} + \sqrt{\rho_R^*}}\]

Note that the denominators are the same as is one of the coefficients in each term in the numerator.

Lastly we compute the energy (the minus and plus correspond to the L and R side respectively)

\[E_k^{**} = E_k^* \mp \sqrt{\rho_k^*} \left( \vec{v}_k^* \cdot \vec{B}_k^* - \vec{v}_k^{**} \cdot \vec{B}_k^{**} \right) \text{sign}\left( B_x \right)\]

Conclusion

And that’s it. We’re done! As you can see the HLLD solver is significantly more complex than the HLLC solver to account for the extra waves. Note that many of the star state primitive variables are used in a bunch of different places so assigning them to member variables is probably a good idea if you’re implementing this as a class.

This post is licensed under CC BY 4.0 by the author.