  This is the second part to our companion paper [18]. Herein, we generalize to
two space dimensions the C-method developed in [20,18] for adding localized,
space-time smooth artificial viscosity to nonlinear systems of conservation
laws that propagate shock waves, rarefaction waves, and contact
discontinuities. For gas dynamics, the C-method couples the Euler equations to
scalar reaction-diffusion equations, which we call C-equations, whose solutions
serve as space-time smooth artificial viscosity indicators for shocks and
contacts. We develop a high-order numerical algorithm for gas dynamics in 2-D
which can accurately simulate the Rayleigh-Taylor (RT) instability with
Kelvin-Helmholtz (KH) roll-up of the contact discontinuity, as well as shock
collision and bounce-back. We implement both directionally isotropic and
anisotropic artificial viscosity schemes, the latter adding diffusion only in
directions tangential to the evolving front. We additionally produce a novel
shock collision indicator function, which naturally activates during shock
collision, and then smoothly deactivates. Moreover, we implement a
high-frequency 2-D wavelet-based noise detector together with an efficient and
localized noise removal algorithm. We provide numerical results for some
classical 2-D test problems, including the RT problem, the Noh problem, a
circular explosion problem from the Liska and Wendroff [13] review paper, the
Sedov blast wave problem, the double Mach 10 reflection test, and a shock-wall
collision problem. In particular, we show that our artificial viscosity method
can eliminate the wall-heating phenomenon for the Noh problem, and thereby
produce an accurate, non-oscillatory solution, even though our simplified
WENO-type scheme fails to run for this problem.
