However, the strong coupling introduced in the equations by the presence of the viscous terms in the definition of relativistic momentum and total energy density required them to treat the difference equations implicitly, which has prevented the development of any multidimensional version of their formulation. They obtained accurate results in the description of strong relativistic shocks with large Lorentz factors in combination with adaptive mesh techniques. In the mid-1980s, Norman and Winkler ( 1986) proposed a new formulation of the difference equations of RHD with an artificial viscosity consistent with the relativistic dynamics of non-perfect fluids. However, despite its popularity, it turned out that Wilson’s approach is unable to accurately describe highly relativistic flows, i.e., with Lorentz factors larger than 2 (see, e.g., Centrella and Wilson, 1984). Almost all numerical codes developed for both special and general RHD in the 1980s (Piran, 1980 Stark and Piran, 1987 Nakamura et al., 1980 Nakamura, 1981 Nakamura and Sato, 1982 Evans, 1986) were based on Wilson’s approach. The code relied on artificial viscosity techniques to handle shock waves and was widely used in numerical cosmology, studies of axisymmetric relativistic stellar collapse, and accretion onto compact objects. Wilson ( 1972, 1979) and collaborators (Centrella and Wilson, 1984 Hawley et al., 1984) made the first attempt to solve the RHD equations in more than one spatial dimension using an Eulerian explicit finite-difference code with monotonic transport. Developments in numerical RHD prior to the year 2003 are reviewed in Martí and Müller ( 2003) and are summarized here for completeness. This review summarizes the progress in grid-based methods for numerical (special) relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) and discusses their application to astrophysical flow. Moreover, in most of these scenarios dynamically important magnetic fields are encountered. The relativistic jets and outflows found in, e.g., micro-quasars, radio-loud AGN and GRB involve flows at relativistic speeds, too. Among these phenomena are core collapse supernovae, X-ray binaries, pulsars, coalescing neutron stars, formation of black holes, active galactic nuclei (AGN) and gamma-ray bursts (GRB). Relativity is a necessary ingredient for describing astrophysical phenomena involving compact objects and flows near the speed of light.
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