

4.1 Van Putten’s
approach
Relying on a formulation of Maxwell’s equations as a hyperbolic
system in divergence form, van Putten [285] has devised a numerical
method to solve the equations of relativistic ideal MHD in flat
spacetime [287
]. Here we only
discuss the basic principles of the method in one spatial
dimension. In van Putten’s approach, the state vector
and the fluxes
of the conservation laws are decomposed
into a spatially constant mean (subscript 0) and a spatially
dependent variational (subscript 1) part,
The RMHD 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 (53). 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 [286
] 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 (
) [288
, 291
].

