In the context of the advection equation of the form $$\frac{\partial \phi}{\partial t}=-u\frac{\partial \phi}{\partial x},$$ the author in [1] discussed how many schemes are based on second-order time averaging, particularly for spatially second-order-accurate methods. In doing so, the author provides an example stating that if $\hat{\phi}_f$ (the subscript $f$ refers to face) is the instantaneous face value, then the time-averaged face value over $\Delta t$ is $$ \phi_f=\phi^*_f=\frac{1}{\Delta t}\int_0^{\Delta t}\hat{\phi}_f(\tau){d}\tau=\phi^n_f-\frac{u \Delta t}{2}\left(\frac{\partial \phi}{\partial x}\right)^n_f, $$ which the author states can be straightforwardly integrated. The asterisk refers to the time average and the superscript refers to the value at time-level $n$. I was not able to reproduce this calculation and was not sure what manipulations occurred to arrive at the last equality.
Context: For more context, a few definitions and equations referred to in the text are needed. The text assumes a control volume over a uniform grid with faces left, $l$, at $-\Delta x/2$, and right, $r$, at $\Delta x/2$ from the cell center. The difference scheme at a cell $i$ is given by $$\phi^{n+1}_i=\phi^n_i-c(\phi_r-\phi_l),$$ where $c$ is the Courant number $u\Delta t/\Delta x$. A few important relations are mentioned. Namely, writing the advective algorithm for constant $u$ in 1D as $$\phi^{n+1}_i=\phi(x,\Delta t)=\phi(x-u\Delta t,0)=\phi^n(x-u\Delta t).$$ Writing $\phi^n(x)$ as a function of the normalized local coordinate, $\xi=(x-x_i)/\Delta x$ gives $$\phi^n(x)=\phi^n_i+f(\xi),$$ where the function $f$ is the homogeneous interpolation function. For conservative schemes, this leads to the relation $$\phi^{n+1}_i=\phi^n_i-c[\phi_r(c)-\phi_l(c)],$$ where face values are functions of the Courant number.
My attempt: What I arrive at with a straight-forward integration is a first order approximation of the time derivative $$\frac{1}{\Delta t}\int_0^{\Delta t}\hat{\phi}_f(\tau){d}\tau=\frac{\phi_f(\Delta t)-\phi_f(0)}{\Delta t},$$ which leads me nowhere. From the other side, the resulting term from the paper looks like the first two terms of the Taylor expansion of $$\phi_f(\Delta t/2)=\phi_f(0)+\frac{\Delta t}{2}\frac{\partial \phi_f(0)}{\partial t}+O(\Delta t^2)=\phi_f(0)-\frac{u\Delta t}{2}\frac{\partial \phi_f(0)}{\partial x}+O(\Delta t^2).$$ However, it is unclear how this term can result from the integral, so I am thinking it is not the right path.
Reference: [1] Leonard, B. P. ``The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection.'' Computer methods in applied mechanics and engineering 88.1 (1991): 17-74. https://doi.org/10.1016/0045-7825(91)90232-U
This is simply using the Taylor series expansion near $t \approx 0$ $$ \frac{1}{\Delta t} \int_0^{\Delta t} \phi(\tau) \ d \tau \approx \frac{1}{\Delta t} \int_0^{\Delta t} \phi(0) + t \phi_\tau(0) \ d\tau = \phi(0) + \frac{\Delta t}{2} \phi_\tau(0) = \phi(0) - \frac{u \Delta t}{2} \frac{\partial \phi}{\partial x}. $$