  The quasi-static code WAKE [P. Mora and T. Antonsen, Phys. Plasmas {\bf 4},
217(1997)] is upgraded to model the propagation of an ultra-relativistic
charged particle beam through a warm background plasma in plasma wakefield
acceleration. The upgraded code is benchmarked against the full
particle-in-cell code OSIRIS [Hemker et al., Phys. Rev. ST Accel. Beams {\bf
3}, 061301(2000)] and the quasi-static code QuickPIC [Huang et al., J. Comp.
Phys. {\bf 217}, 658 (2006)]. The effect of non-zero plasma temperature on the
peak accelerating electric field is studied for a two bunch electron beam
driver with parameters corresponding to the plasma wakefield acceleration
experiments at FACET. It is shown that plasma temperature does not affect the
energy gain and spread of the accelerated particles despite suppressing the
peak accelerating electric field. The role of plasma temperature in improving
the numerical convergence of the electric field with the grid resolution is
discussed.
