  The aim of this paper is to develop a general method for constructing
approximation schemes for viscosity solutions of fully nonlinear pathwise
stochastic partial differential equations, and for proving their convergence.
Our results apply to approximations such as explicit finite difference schemes
and Trotter-Kato type mixing formulas. The irregular time dependence disrupts
the usual methods from the classical viscosity theory for creating schemes that
are both monotone and convergent, an obstacle that cannot be overcome by
incorporating higher order correction terms, as is done for numerical
approximations of stochastic or rough ordinary differential equations. The
novelty here is to regularize those driving paths with non-trivial quadratic
variation in order to guarantee both monotonicity and convergence.
  We present qualitative and quantitative results, the former covering a wide
variety of schemes for second-order equations. An error estimate is established
in the Hamilton-Jacobi case, its merit being that it depends on the path only
through the modulus of continuity, and not on the derivatives or total
variation. As a result, it is possible to choose a regularization of the path
so as to obtain efficient rates of convergence. This is demonstrated in the
specific setting of equations with multiplicative white noise in time, in which
case the convergence holds with probability one. We also present an example
using scaled random walks that exhibits convergence in distribution.
