SEMINAR: Limiting Techniques for High-Order Finite Element Discretizations of Conservation Laws

Abstract:

The main focus of this talk is on the design of high-resolution finite
element schemes for scalar conservation laws and hyperbolic systems.
A unified limiting strategy is proposed for continuous and discontinuous
Galerkin approximations. Using the Bernstein basis representation of
polynomial shape functions, we modify the high-order discretization
so as to enforce local discrete maximum principles for the coefficients
and guarantee that numerical solutions stay in the admissible range.

The development of accuracy-preserving limiters for high-order Bernstein
finite elements requires careful revisions of algorithms designed for
linear and multilinear Lagrange elements. The proposed element-based
extensions of algebraic flux correction tools include

(i) a new low-order upwinding strategy and high-order variational
stabilization for continuous finite elements,

(ii) localized limiters for antidiffusive element contributions, and

(iii) an accuracy-preserving smoothness indicator that allows violations
of discrete maximum principles at smooth peaks.

Extensions of element-based limiters to nonlinear hyperbolic systems
will be presented in the context of piecewise-linear finite element
approximations to the Euler equations of gas dynamics.

This is joint work with R. Anderson, V. Dobrev, Tz. Kolev, C. Lohmann,
M. Quezada de Luna, S. Mabuza, R. Rieben, J.N. Shadid, and V. Tomov