  The numerical approximation of the semilinear Klein--Gordon equation in the
$d$-dimensional space, with $d=1,2,3$, is studied by analyzing the consistency
errors in approximating the solution. By discovering and utilizing a new
cancellation structure in the semilinear Klein--Gordon equation, a
low-regularity correction of the Lie splitting method is constructed, which can
have second-order convergence in the energy space under the regularity
condition $(u,\partial_tu)\in L^\infty(0,T;H^{1+\frac{d}{4}}\times
H^{\frac{d}{4}})$, where $d=1,2,3$ denotes the dimension of space. In one
dimension, the proposed method is shown to have a convergence order arbitrarily
close to $\frac53$ in the energy space for solutions in the same space, i.e. no
additional regularity in the solution is required. Rigorous error estimates are
presented for a fully discrete spectral method with the proposed low-regularity
time-stepping scheme. Numerical examples are provided to support the
theoretical analysis and to illustrate the performance of the proposed method
in approximating both nonsmooth and smooth solutions of the semilinear
Klein--Gordon equation.
