Numerical Schemes for the (Real Gas) MHD Equations

In recent years Godunov-type schemes have become very popular in numerical gas dynamics. For simulations in two and three space dimensions the solution to one-dimensional Riemann problems has to be approximated at cell interfaces. Therefore a major step in the development of this type of schemes is the construction of a good approximation to the Riemann problem. We have developed a new Riemann solver for the ideal MHD equations and have compared it with different solution strategies from the literature.

In many applications as for example in the solar photosphere, the gas pressure is not given by the equation of state for a perfect gas. In this case the solution to the Riemann problem can become far more complicated; compound wave can now arise even in the purely hydrodynamic case. Therefore the solution method has to be modified to cope with the different thermodynamic setting. We have again compared different methods for the Euler equations of gas dynamics and extended them to the real gas MHD equations.

Approximate Riemann Solvers for the Ideal Gas MHD Equations

We implemented and tested six approximate Riemann solvers for the Ideal Gas MHD equations. Five solvers are taken from the literature: Bell-Colella-Trangenstein (BCT), Dai-Woodward (DW), Harten-Lax-van Leer-Einfeldt (HLLE), Lax-Friedrichs (LF), and Roe. The sixth solver is our new MHD-HLLEM approximate Riemann solver which generalizes ideas of the Euler-HLLEM scheme to MHD. We find that our new solver incorporates very little numerical dissipation; at the same time it is still very robust. In all our tests MHD-HLLEM outperforms the other five solvers in the sense that it reaches the same errors in significantly less computational time. A particularly striking example is the advection of a smooth impulse in the magnetic field in 2d which is included below.

A standard technique for enhancing the efficiency of a code is the extension of a first-order scheme to (formal) second-order accuracy by means of limited linear reconstruction in space and a two-step Runge-Kutta time integration. In one space dimension realizing this approach is straightforward, whereas on unstructured triangular grids we were not able to perform the Rayleigh-Taylor simulation with the limiters commonly used. Our new limiter DEOmod cures these problems, and it also leads to more efficient schemes than the standard approaches.

Example: Advection of a Smooth Magnetic Impulse in 2d

Results of Different Solvers on the Same Grid:

(Click to see a 430 kB animated GIF)

Efficiencies (i.e., Error vs. Cputime) of Different Solvers:

Related Publications:

Approximate Riemann Solvers for Real Gasdynamics

Riemann Solvers

For simulations in the solar photosphere the assumption of a perfect gas law is not appropriate: The plasma in this region is partially ionized so that the mean molecular weight, which is required for the computation of the pressure, is no longer constant. This leads to a complicated relationship between the pressure, the density, and the internal energy of the plasma. In our Riemann solver based schemes for the perfect gas MHD equations the simple structure of the pressure law was used to derive a compact form of the eigenvalues and eigenvectors of the flux Jacobian. Therefore all the schemes (with the exception of the very diffusive Lax-Friedrichs scheme) cannot be directly used for the computations in the solar photosphere. We have extended two approaches suggested for the Euler equations of gas dynamics to generalize solvers for perfect gases to the real gas case. Both approaches lead to very similar results and both resolve complicated compound wave structures in solutions to the Riemann problem accurately.

MHD Riemann problem for a van der Waals gas:
Reference solution (DW scheme, 10000 points) and DW/Lax-Friedrichs solutions with 800 points).

Tabularized Equation of State

In one time step the solvers we have constructed for the MHD equations with a general equation of state have to compute the pressure from the conservative quantities at least once in each cell. Some solvers do also require the computation of the internal energy from the pressure and the density. In our case of a partially-ionized plasma these conversions require costly iterative procedures. For computational efficiency it was therefore necessary to reduce the number of calls to the exact gas law considerably. We achieved this aim by tabularizing the pressure and the internal energy and using bilinear interpolation: If we allow for 1.5% deviation from the exact values, the use of the table saves up to 90% of computational time. On the other hand, for this level of accuracy a table with about 25 million entries is required which occupies more than 2 Gigabytes of memory. Therefore we now use a locally-adapted table which saves up to 70% of memory.
Note that quite simple tests show that the resolution of the table has to be coupled to the spatial resolution. Otherwise no grid convergence is obtained.

Numerical study of the connection between grid spacing h = 2/N and table resolution K.

Close-up of the velocity


 
Grid convergence rates

Required levels of an adaptive table in a MHD application with partially-ionized plasma.

Related Publications: