  In this first part of two papers, we extend the C-method developed in [40]
for adding localized, space-time smooth artificial viscosity to nonlinear
systems of conservation laws that propagate shock waves, rarefaction waves, and
contact discontinuities in one space dimension. For gas dynamics, the C-method
couples the Euler equations to a scalar reaction-diffusion equation, whose
solution $C$ serves as a space-time smooth artificial viscosity indicator.
  The purpose of this paper is the development of a high-order numerical
algorithm for shock-wall collision and bounce-back. Specifically, we generalize
the original C-method by adding a new collision indicator, which naturally
activates during shock-wall collision. Additionally, we implement a new
high-frequency wavelet-based noise detector together with an efficient and
localized noise removal algorithm. To test the methodology, we use a highly
simplified WENO-based discretization scheme. We show that our scheme improves
the order of accuracy of our WENO algorithm, handles extremely strong
discontinuities (ranging up to nine orders of magnitude), allows for shock
collision and bounce back, and removes high frequency noise. The causes of the
well-known "wall heating" phenomenon are discussed, and we demonstrate that
this particular pathology can be effectively treated in the framework of the
C-method. This method is generalized to two space dimensions in the second part
of this work [41].
