Van Putten’s approach

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 $u$ and the fluxes $F$ 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 $∗ u1$ , which reads

together with the conservation condition

The quantities $u1 ∗$ 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 $u (t,x)$ is then obtained from $∗ u1 (t,x)$ 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 ( $<$ 4.25) [288 , 291 ].