The standard pseudospectral algorithm for simulating homogeneous turbulent shear flow by Rogallo (1981) solves the governing equations on a mesh in that deforms with the mean shear in physical space. The deformation allows the velocity and pressure to be decomposed using a traditional Fourier series. To limit the maximum degree of deformation, the Rogallo algorithm grid periodically maps the variables on a positively deformed mesh onto a mesh that is deformed in the opposite direction. This step is known as "remeshing." As a consequence of the stretching of the coordinate that accompanies the mesh deformation, 50% of the wavenumbers in the direction of the shear must be zeroed out. This leads to a sudden decrease in the turbulent kinetic energy and energy dissipation rate. Here we present an alternate pseudospectral algorithm for homogeneous turbulent shear flow that avoids the need to periodically remesh the coordinate in physical space. The algorithm avoids remeshing by performing a phase shift to maintain the physical space representation of all variables on the orthogonal mesh throughout the calculation. Comparisons with the traditional Rogallo algorithm show significant improvements, particularly at high shear rates, where frequent remeshing steps leads to significant losses in the dissipation rate. With the new algorithm, we explore the dependence of small-scale turbulence statistics such as strain and vorticity on the shear parameter.