  Local gyrokinetic simulations solve the gyrokinetic equations with
homogeneous background gradients, typically using a doubly periodic domain in
the (x,y) plane (i.e. perpendicular to the field line). Spatial Fourier
representations are almost universal in local gyrokinetic codes, and the
wavevector-remap method was introduced in [Hammett et. al., Bull Am Phys Soc
VP1 136, (2006)] as a simple method for expressing the local gyrokinetic
equations with a background shear flow in a Fourier representation. Although
extensively applied, the wavevector-remap method has not been formally shown to
converge, and suffers from known unphysicality when the solutions are plotted
in real space [Fox et. al. PPCF 59, 044008]. In this work, we use an analytic
solution in slab geometry to demonstrate that wavevector-remap leads to
incorrect smeared non-linear coupling between modes. We derive a correct,
relatively simple method for solving local gyrokinetics in Fourier space with a
background shear flow, and compare this to the wavevector-remap method. This
allows us to show that the error in wavevector-remap can be seen as an
incorrect rounding in wavenumber space in the nonlinear term. By making minor
modifications to the nonlinear term, we implement the corrected
wavevector-remap scheme in the GENE[25] code and compare results of the
original and corrected wavevector-remap for standard nonlinear benchmark cases.
Certain physical phenomena are impacted by the errors in the original remap
scheme, and these numerical artefacts do not reduce as system size increases:
that is, original wavevector-remap scheme does not converge to the correct
result.
