In his 1954 dissertation, Godunov proved that monotone linear numerical schemes for solving partial differential equations can be at most first-order accurate. Consequently, a number of nonlinear numerical schemes have been proposed with the hopes of obtaining monotonicity. In the finite element community, residual-based artificial viscosities have traditionally been added to a stabilized formulation. The design of these discontinuity-capturing terms has been largely motivated by entropy analysis, and their implementation is often more of an art than a science.
In this work, we develop an alternative approach to the design of discontinuity-capturing terms through the framework of the variational multiscale (VMS) method. In the VMS method, the solution is decomposed into a coarse-scale component, which we aim to resolve numerically using a finite element method, and a fine-scale component. The scale splitting is defined by means of an optimality condition. The goal of VMS is to model the influence of the fine-scales on the coarse-scales analytically using this construct. To ensure a monotone solution, we subject our optimality condition to a total variation constraint on the coarse-scale component. This definition leads to a variational formulation of a character altogether different than other VMS schemes. In fact, it leads to a multiscale finite element formulation with a new pair of discontinuity-capturing terms. We analyze the structure of this multiscale formulation and compare it to more traditional shock-capturing schemes.