# A positive conservative method for magnetohydrodynamics based
on HLL and Roe methods

### P. Janhunen

#### J. Comp. Phys., accepted

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*Abstract.*
The exact Riemann problem solutions of the usual equations of ideal
magnetohydrodynamics (MHD) can have negative pressures, if the initial
data has $\nabla\cdot{\bf B}\ne 0$. This creates a problem for
numerical solving because in a first order finite-volume conservative
Godunov-type method one cannot avoid jumps in the normal magnetic
field component even if the magnetic field was divergenceless in the
three-dimensional sense. We show that by allowing magnetic monopoles
in MHD equations and properly taking into account the magnetostatic
contribution to the Lorentz force, an additional source term appears
in Faraday's law only. Using the Harten-Lax-vanLeer-Einfeldt (HLL)
Riemann solver and discretizing the source term in a specific manner,
we obtain a method which is positive and conservative. We show
positivity by extensive numerical experimentation. This MHD-HLL
method is positive and conservative but rather diffusive, thus we show
how to hybridize this method with the Roe method to obtain a much
higher accuracy while still retaining positivity. The result is a
fully robust positive conservative scheme for ideal MHD, whose
accuracy and efficiency properties are similar to the first order Roe
method and which keeps $\nabla\cdot{\bf B}$ small in the same sense as
Powell's method. As a special case, a method with similar
characteristics for accuracy and robustness is obtained for the Euler
equations as well.