**2. A Compact Difference Scheme**

A linearized numerical method that combines the super-convergence approximation *L*<sup>2</sup> − 1*<sup>σ</sup>* with the order reduction method is derived. Some further notations are fixed before we continue. Take two positive integers *M* and *n*0, let *h* = *<sup>L</sup> <sup>M</sup>* , *<sup>τ</sup>* <sup>=</sup> *<sup>s</sup> <sup>n</sup>*<sup>0</sup> and denote *xi* = *i h* for *<sup>i</sup>* = 0, ... , *<sup>M</sup>*; *tk* = *<sup>k</sup> <sup>τ</sup>* and *tk*<sup>+</sup>*<sup>σ</sup>* = (*k* + *σ*) *τ*, for *k* = −*n*0, ... , *N*, where *N* = " *T τ* # . Using the points *xi* in space and *tk* in time, we cover the space-time domain by Ω*h<sup>τ</sup>* = Ω*<sup>h</sup>* × Ω*τ*, where Ω*<sup>h</sup>* = {*xi* | 0 ≤ *i* ≤ *M*} and Ω*<sup>τ</sup>* = {*tk* | −*n*<sup>0</sup> ≤ *k* ≤ *N*}.
