  This work presents the design of nonlinear stabilization techniques for the
finite element discretization of Euler equations in both steady and transient
form. Implicit time integration is used in the case of the transient form. A
differentiable local bounds preserving method has been developed, which
combines a Rusanov artificial diffusion operator and a differentiable shock
detector. Nonlinear stabilization schemes are usually stiff and highly
nonlinear. This issue is mitigated by the differentiability properties of the
proposed method. Moreover, in order to further improve the nonlinear
convergence, we also propose a continuation method for a subset of the
stabilization parameters. The resulting method has been successfully applied to
steady and transient problems with complex shock patterns. Numerical
experiments show that it is able to provide sharp and well resolved shocks. The
importance of the differentiability is assessed by comparing the new scheme
with its non-differentiable counterpart. Numerical experiments suggest that,
for up to moderate nonlinear tolerances, the method exhibits improved
robustness and nonlinear convergence behavior for steady problems. In the case
of transient problem, we also observe a reduction in the computational cost.
