The RMHD (for relativistic MHD) equations then become a system of evolution equations for the integrated variational parts , which reads

together with the conservation condition

The quantities are defined as

They are continuous, and standard methods can be used to integrate the system (44). Van Putten uses a leapfrog method.

The new state vector is then obtained from by numerical differentiation. This process can lead to oscillations in the case of strong shocks and a smoothing algorithm should be applied. Details of this smoothing algorithm and of the numerical method in one and two spatial dimensions can be found in [180] together with results on a large variety of tests.

Van Putten has applied his method to simulate relativistic hydrodynamic and magneto hydrodynamic jets with moderate flow Lorentz factors (< 4.25) [182, 184].

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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