4.2 Relativistic SPH4 Other Developments4 Other Developments

4.1 Van Putten's approach 

Relying on a formulation of Maxwell's equations as a hyperbolic system in divergence form, van Putten [179] has devised a numerical method to solve the equations of relativistic ideal MHD in flat spacetime [181Jump To The Next Citation Point In The Article]. Here we only discuss the basic principles of the method in one spatial dimension. In van Putten's approach, the state vector tex2html_wrap_inline5699 and the fluxes tex2html_wrap_inline5963 of the conservation laws are decomposed into a spatially constant mean (subscript 0) and a spatially dependent variational (subscript 1) part

equation762

The RMHD (for relativistic MHD) equations then become a system of evolution equations for the integrated variational parts tex2html_wrap_inline6061, which reads

  equation771

together with the conservation condition

equation777

The quantities tex2html_wrap_inline6061 are defined as

equation782

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

The new state vector tex2html_wrap_inline6065 is then obtained from tex2html_wrap_inline6067 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 [180Jump To The Next Citation Point In The Article] 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) [182Jump To The Next Citation Point In The Article, 184Jump To The Next Citation Point In The Article].



4.2 Relativistic SPH4 Other Developments4 Other Developments

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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