  We perform a numerical simulation of quantum turbulence produced by thermal
counterflow in superfluid $^4$He by using the vortex filament model with the
full Biot--Savart law. The pioneering work of Schwarz has two shortcomings: it
neglects the non-local terms of the Biot--Savart integral (known as the
localized induction approximation, LIA) and it employs an unphysical mixing
procedure to sustain the statistically steady state of turbulence. For the
first time we have succeeded in generating the statistically steady state under
periodic boundary conditions without using the LIA or the mixing procedure.
This state exhibits the characteristic relation $L=\gamma^2 v_{ns}^2$ between
the line-length density $L$ and the counterflow relative velocity $v_{ns}$ and
there is quantitative agreement between the coefficient $\gamma$ and some
measured values. The parameter $\gamma$ and some anisotropy parameters are
calculated as functions of temperature and the counterflow relative velocity.
The numerical results obtained using the full Biot--Savart law are compared
with those obtained using the LIA. The LIA calculation constructs a layered
structure of vortices and does not proceed to a turbulent state but rather to
another anisotropic vortex state; thus, the LIA is not suitable for simulations
of turbulence.
