Finite Element Approximation of Nonlinear Systems Developing Shocks, Fronts and Interfaces
Abstract
The objective of the proposed research is to investigate the stabilization of time-dependent nonlinear conservation equations by using entropy viscosity. It will develop methods that guarantee some sort of maximum principle (or invariant domain property for systems), incorporate entropy dissipation, run with optimal CFL, work on arbitrary meshes in any space dimension, generate high-order accuracy for smooth solutions, and be easily parallelizable. This project will extend the continuous finite element method to at least third-order accurate in space. This will require the development of a new two-level technique to blend a first-order method (which will be used to enforce the maximum principle bounds) and a high-order method (entropy viscosity). The blending will be done by using a FCT-like limitation technique. The scalar results will be extended to nonlinear hyperbolic systems such as Euler compressible flows and Shallow water equations. This will be done without invoking Riemann solvers, either approximate or exact. The key difficulty here is that the standard notion of maximum principle is not valid. However, most physical systems have positively invariant domains (like positive density, internal energy, and more, for the Euler equations); these will be exploited to preserve invariant domains.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jan 12, 2017
- Source ID
- W911NF1510517
Entities
People
- Jean-luc Guermond
Organizations
- Army Contracting Command
- Texas A&M University
- United States Army