The coherent
synchrotron radiation (CSR) is one problem limiting the performance
of high intensity electron accelerators. The complexity of the physical
mechanisms underlying the onset of instabilities due to CSR demands
for reliable simulation codes. In the past, codes based on Lie algebraic
techniques have been very efficient to treat transport problems in accelerators.
The extension of these methods to the non-linear case is ideally suited
to treat CSR instability problems. We report on the development of a
numerical code, based on the solution of the Vlasov equation, with the
inclusion of non-linear contribution due to wake field effects. The
proposed solution method exploits an algebraic technique, using exponential
operators. We show that the integration procedure is capable of reproducing
the onset of an instability and the effects associated with bunching
mechanisms leading to the growth of the instability itself. In addition,
considerations on the threshold of the instability for Gaussian bunches
and a preliminary study of CSR effects on SPARXINO are also reported.