  We consider a semilinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB)
equation with singularity which is related to a stochastic control problem with
fuel constraint. The fuel constraint translates into a singular initial
condition for the HJB equation. We first propose a transformation based on a
change of variables that gives rise to an equivalent HJB equation with
nonsingular initial condition but irregular coefficients. We then construct
explicit and implicit numerical schemes for solving the transformed HJB
equation and prove their convergences by establishing an extension to the
result of Barles and Souganidis (1991).
