  An efficient and compact approach to the inclusion of dissipative effects in
Non-Equilibrium Green's Function (NEGF) simulations of electronic systems is
introduced. The algorithm is based on two well known methods in the literature,
firstly that of the so-called Recursive Green's Function (RGF) and secondly
that of B\"uttiker probes. Numerical methods for exact evaluation of the
Jacobian are presented by a direct extension to RGF which can be modularly
included in any codebase that uses it presently. Then using both physical
observations and numerical methods, the computation time of the B\"uttiker
probe Jacobian is improved significantly. An improvement to existing phonon
models within B\"uttiker probes is then demonstrated in the simulation of fully
atomistic graphene nanoribbon based field effect transistors in n-i-n and p-i-n
operation.
