  The quasilinear theory describes the resonant interaction between particles
and waves with two coupled equations: one for the evolution of the particle
probability density function(\textit{pdf}), the other for the wave spectral
energy density(\textit{sed}). In this paper, we propose a conservative Galerkin
scheme for the quasilinear model in three-dimensional momentum space and
three-dimensional spectral space, with cylindrical symmetry.
  We construct an unconditionally conservative weak form, and propose a
discretization that preserves the unconditional conservation property, by
"unconditional" we mean that conservation is independent of the singular
transition probability. The discrete operators, combined with a consistent
quadrature rule, will preserve all the conservation laws rigorously. The
technique we propose is quite general: it works for both relativistic and
non-relativistic systems, for both magnetized and unmagnetized plasmas, and
even for problems with time-dependent dispersion relations.
  We represent the particle \textit{pdf} by continuous basis functions, and use
discontinuous basis functions for the wave \textit{sed}, thus enabling the
application of a positivity-preserving technique. The marching simplex
algorithm, which was initially designed for computer graphics, is adopted for
numerical integration on the resonance manifold. We introduce a semi-implicit
time discretization, and discuss the stability condition. In addition, we
present numerical examples with a "bump on tail" initial configuration, showing
that the particle-wave interaction results in a strong anisotropic diffusion
effect on the particle \textit{pdf}.
