  We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
$C$-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution $C(x,t)$ is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the $C$-method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our $C$-method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.
