Numerical Analysis Seminar

Date: February 20, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 220

Speaker: Dmitri Kuzmin, Dortmund University of Technology

Title: Bounds-preserving limiters for continuous high-order finite element discretizations of hyperbolic conservation laws

Abstract: In this talk, we constrain high-order finite element approximations to hyperbolic conservation laws using localized corrections to enforce discrete maximum principles. The use of Bernstein basis functions ensures that numerical solutions stay in the admissible range. The design of accuracy-preserving FCT schemes for high-order Bernstein finite elements requires a major revision of algorithms designed for low-order Lagrange elements. In this talk, we discretize the linear advection equation using an element-based FCT algorithm which features: (i) a new discrete upwinding strategy leading to variation diminishing low-order approximations with compact stencils, (ii) a high-order stabilization operator based on the divergence of the difference between two gradient approximations, (iii) localized limiters for antidiffusive element contributions, and (iv) an accuracy-preserving smoothness indicator that allows violations of strict maximum principles at smooth peaks. Additionally, we present limiters that constrain artificial diffusion coefficients or the difference between finite element basis functions corresponding to high-order and piecewise-linear approximations. Extensions of FCT to hyperbolic systems will also be discussed. This is joint work with C. Lohmann, J.N. Shadid, S. Mabuza, and Manuel Quezada de Luna