  We will introduce Euler-Maruyama approximations given by an orthogonal system
in $L^{2}[0,1]$ for high dimensional SDEs, which could be finite dimensional
approximations of SPDEs. In general, the higher the dimension is, the more one
needs to generate uniform random numbers at every time step in numerical
simulation. The schemes proposed in this paper, in contrast, can deal with this
problem by generating very few uniform random numbers at every time step. The
schemes save time in the simulation of very high dimensional SDEs. In
particular, we conclude that an Euler-Maruyama approximation based on the Walsh
system is efficient in high dimensions.
