Take the linearised FKPP equation {partial_{t}h = partial^{2}_{x}h + h} with boundary condition h(m(t), t) = 0. Depending on the behaviour of the initial condition h0(x) = h(x, 0) we obtain the asymptotics—up to a o(1) term r(t)—of the absorbing boundary m(t) such that {omega(x) := lim_{ttoinfty} h(x + m(t) ,t)} exists and is non-trivial. In particular, as in Bramson’s results for the non-linear FKPP equation, we recover the celebrated {-3/2log t} correction for initial conditions decaying faster than {x^{nu}e^{-x}} for some {nu < -2}. Furthermore, when we are in this regime, the main result of the present work is the identification (to first order) of the r(t) term, which ensures the fastest convergence to {omega(x)}. When h0(x) decays faster than {x^{nu}e^{-x}} for some {nu < -3}, we show that r(t) must be chosen to be {-3sqrt{pi/t}}, which is precisely the term predicted heuristically by Ebert–van Saarloos (Phys. D Nonlin. Phenom. 146(1): 1–99, 2000) in the non-linear case (see also Mueller and Munier Phys Rev E 90(4):042143, 2014, Henderson, Commun Math Sci14(4):973–985, 2016, Brunet and Derrida Stat Phys 1-20, 2015). When the initial condition decays as {x^{nu}e^{-x}} for some {nu in [-3, -2)}, we show that even though we are still in the regime where Bramson’s correction is {-3/2 log t}, the Ebert–van Saarloos correction has to be modified. Similar results were recently obtained by Henderson CommunMathSci 14(4):973–985, 2016 using an analytical approach and only for compactly supported initial conditions.