  The paper proposes a way to control the viscosity of numerical approximation
in the contact SPH method. This variant of SPH contains momentum and energy
fluxes in the right-hand sides of the equations, which are calculated using the
solution of the Riemann problem between each pair of neighboring particles
within the smoothing kernel support radius, which is similar to the procedure
for calculating fluxes across cell boundaries in Godunov schemes. Such SPH
method does not require the use of artificial viscosity, because the
significant numerical viscosity is already introduced by a Riemann problem
solution. The magnitude of numerical viscosity decreases linearly with particle
size, however, it becomes comparable with the physical viscosity for most
materials when the particle size is about $\thicksim 1\,$nm, which hampers the
correct accounting for viscous effects in real-life problems. In this study we
develop a method for reducing the viscous stresses of numerical origin, for
which a correcting viscous stress tensor is constructed on the basis of the
analytical solution for discontinuous viscous flow. The use of such correction
makes it possible to improve the agreement with the experiment in the
simulation of viscous flows.
