  A fully discrete approximation of the one-dimensional stochastic heat
equation driven by multiplicative space-time white noise is presented. The
standard finite difference approximation is used in space and a stochastic
exponential method is used for the temporal approximation. Observe that the
proposed exponential scheme does not suffer from any kind of CFL-type step size
restriction. When the drift term and the diffusion coefficient are assumed to
be globally Lipschitz, this explicit time integrator allows for error bounds in
$L^q(\Omega)$, for all $q\geq2$, improving some existing results in the
literature. On top of this, we also prove almost sure convergence of the
numerical scheme. In the case of non-globally Lipschitz coefficients, we
provide sufficient conditions under which the numerical solution converges in
probability to the exact solution. Numerical experiments are presented to
illustrate the theoretical results.
