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 [287Jump To The Next Citation Point]. 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,
u(t,x) = u0(t) + u1(t,x), F (t,x ) = F0(t) + F1 (t,x). (52 )
The RMHD equations then become a system of evolution equations for the integrated variational parts ∗ u1, which reads
∂u1∗ -----+ F1 = 0, (53 ) ∂t
together with the conservation condition
dF0-= 0. (54 ) dt
The quantities u1 ∗ are defined as
∫ x u1 ∗(t,x) = u1 (t,y )dy. (55 )
They are continuous, and standard methods can be used to integrate the system (53View Equation). 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 [286Jump To The Next Citation Point] 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) [288Jump To The Next Citation Point291Jump To The Next Citation Point].

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