Discontinuity Capturing and the Variational Multiscale Method
John Evans, University of Texas
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.
Abstract Author(s): John Evans, Thomas J.R. Hughes, Giancarlo Sangalli