  We develop a new model-independent method to probe the constancy of the speed
of light $c$. In our method, the degeneracy between the cosmic curvature and
the speed of light can be eliminated, which makes the test more natural and
general. Combining the independent observations of Hubble parameter $H(z)$ and
luminosity distance $d_L(z)$, we use the model-independent smoothing technique,
Gaussian processes, to reconstruct them and then detect variation of the speed
of light. We find no signal of deviation from the present value of the speed of
light $c_0$. Moreover, to demonstrate the improvement in probing the constancy
of the speed of light from future experiments, we produce a series of simulated
data. The Dark Energy Survey will be able to detect $\Delta c /c_0 \sim 1\%$ at
$\sim 1.5\sigma$ confidence level and $\Delta c /c_0 \sim 2\%$ at $\sim
3\sigma$ confidence level. If the errors are reduced to one-tenth of the
expected DES ones, it can detect a $\Delta c /c_0 \sim 0.1\%$ variation at
$\sim 2\sigma$ confidence level.
