We propose a modification for the tensor of artificial viscosity employable for generally comoving, curvilinear grids. We present a strong conservation form for the equations of radiation hydrodynamics for studying nonlinear pulsations of stars. However, the modification we propose is of general mathematical nature. We study a differential geometrically consistent artificial viscosity analytically and visualize a comparison of our approach to previous implementations by applying it to a simple self-similar velocity field which has a direct application in stars as the fundamental mode of pulsation is radial. We first give a general introduction to artificial viscosity and motivate its application in numerical computations. We then show how a tensor of artificial viscosity has to be designed when going beyond common static Eulerian or Lagrangian comoving rectangular grids. We derive and state the modified equations which include metrical terms that adjust the isotropic (pressure) part of the tensor of artificial viscosity.