We study the asymptotic behavior of solutions to the viscous
Burgers equation by presenting a new asymptotic approximate
solution. This approximate solution, called a diffusion wave
approximate solution to the viscous Burgers equation of $k$-th
order, is expanded in terms of the initial moments up to $k$-th
order. Moreover, the spatial and time shifts are introduced
into the leading order term to capture precisely the effect
of the initial data on the long-time behavior of the actual
solution. We also show the optimal convergence order in $L^p$-norm,
$1\leq p\leq \infty$, of the diffusion wave approximate solution
of $k$-th order. These results allow us to obtain the convergence
of any higher order in $L^p$-norm by taking such a diffusion
wave approximate solution with order $k$ large enough.