  We introduce an adaptive viscosity regularization approach for the numerical
solution of systems of nonlinear conservation laws with shock waves. The
approach seeks to solve a sequence of regularized problems consisting of the
system of conservation laws and an additional Helmholtz equation for the
artificial viscosity. We propose a homotopy continuation of the regularization
parameters to minimize the amount of artificial viscosity subject to
positivity-preserving and smoothness constraints on the numerical solution. The
regularization methodology is combined with a mesh adaptation strategy that
identifies the shock location and generates shock-aligned meshes, which allows
to further reduce the amount of artificial dissipation and capture shocks with
increased accuracy. We use the hybridizable discontinuous Galerkin method to
numerically solve the regularized system of conservation laws and the
continuous Galerkin method to solve the Helmholtz equation for the artificial
viscosity. We show that the approach can produce approximate solutions that
converge to the exact solution of the Burgers' equation. Finally, we
demonstrate the performance of the method on inviscid transonic, supersonic,
hypersonic flows in two dimensions. The approach is found to be accurate,
robust and efficient, and yields very sharp yet smooth solutions in a few
homotopy iterations.
