  We investigate various variants of the Hermitean version of the Local Bosonic
Algorithm proposed by M. L\"uscher. The model used is two-dimensional Quantum
Electrodynamics (QED) with two flavours of massive Wilson fermions. The
simplicity of the model allows high statistics simulations close to the chiral
and continuum limit.
  To find optimal CPU cost behaviour, we vary the approximation polynomial
parameters $n$ and \epsilon$ as well as the number of over-relaxation steps
within each trajectory. We find flat behaviour around the optimum and a modest
gain with respect to the Hybrid Monte Carlo algorithm for all variants.
  On the technical side, we demonstrate that a noisy Metropolis acceptance step
is possible also for the Hermitean variant. The numerical instabilities
appearing in the evaluation Chebyshev polynomial are investigated. We propose a
quantitative criterion for these instabilities and a reordering scheme of the
roots reducing the problem.
  The more physical problem of topological charge sectors and metastability is
addressed. We find no plateau in the effective pion mass if metastabilities
become too large.
