Hyperbolic problems can satisfy maximum principles or positivity preservation. In this work, we focus on the development of fully implicit finite ellement methods with nonlinear stabilization based on artificial difusion techniques that keep at the discrete level these interesting properties. The nonlinear viscosity is equal to a nonlinear shock detector times a linear viscosity. The idea is to activate the full linear viscosity on discontinuities/shocks, and switch it off in smooth regions. The shock detector must be such that it takes value 1 on local
extrema (to satisfy discrete maximum principles), and switch off for linear functions, i.e., linearity-preserving (for accuracy purposes). In this sense, we work with shock detectors in the line of [1]. Another ingredient is the expression of the Laplacian term. Following [3], we consider a graph-Laplacian term. Finally, the linear viscosity is edgebased and follows the ideas in flux-corrected transport methods [2]; we consider the minimum amount of viscosity needed to prove monotonicity properties. The resulting scheme satisfy discrete maximum principles and positivity, and is local variation diminishing. Further, following the ideas in [4] we can prove Lipschitz continuity. However, it is not enough to end up with a useful numerical method, since the nonlinear convergence of the resulting algorithms is extremely complicated. In this sense, we have developed a smooth version of the framework, in which we can prove that the resulting nonlinear stabilization term is C2-continuous. The resulting schemes
can still keep the monotonicity properties, but are much cheaper (due to a much better nonlinear convergence). We have considered the Newton method with line search and Anderson acceleration techniques. For a large smoothing parameter, the method is cheaper and more dissipative, whereas for a zero smoothing parameter, we recover the original scheme.