Well-behavedness of standard finite difference schemes for the transformed PDE

Investigate whether standard finite difference schemes (e.g., θ-schemes) are well behaved—i.e., stable and accurate—when applied to the transformed partial differential equation u_τ = (1/2) u_{xx} − 2 (1/f) u_x (equation (5.10)) obtained via the change of variables y = f(x), especially in the presence of a singular drift coefficient near x = 0 and potential convection-dominated regimes.

Background

To address non-uniqueness, the paper considers Çetin’s alternative Cauchy problem and proposes a change of variables y = f(x) (defined by equation (3.5)) that moves the boundary at infinity to x = 0, yielding PDE (5.10). This transformation can introduce a singular drift at x = 0 and produce convection-dominated behavior, which is known to challenge standard finite difference methods.

While existence and uniqueness of the continuous solution are ensured, the numerical behavior (stability and accuracy) of standard discretizations for this transformed PDE has not been established and is flagged by the authors as unclear.