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In this article, we present a state-of-the-art algorithm for solving therelativistic viscous hydrodynamics equation with the QCD equation of state. Thenumerical method is based on the second-order Godunov method and has lessnumerical dissipation, which is crucial in describing of quark-gluon plasma inhigh-energy heavy-ion collisions. We apply the algorithm to several numericaltest problems such as sound wave propagation, shock tube and blast waveproblems. In sound wave propagation, the intrinsic numerical viscosity ismeasured and its explicit expression is shown, which is the second-order ofspatial resolution both in the presence and absence of physical viscosity. Theexpression of the numerical viscosity can be used to determine the maximum cellsize in order to accurately measure the effect of physical viscosity in thenumerical simulation.

A new scheme of causal viscous hydrodynamics for relativistic heavy-ion collisions: A Riemann solver for quark-gluon plasma