Prof. Dr. Florian Frank from the Department of Mathematics, University of Erlangen-Nuremberg, will give a SFB 1313 lecture on Thursday, June 6th. His lecture will be about: “Enforcing discrete maximum principles in discontinuous Galerkin schemes with application to phase-field methods”
Date: June 6, 2019
Time: 2:30 pm
Place: IANS, Pfaffenwaldring 57, Room 7.122
Title: "Enforcing discrete maximum principles in discontinuous Galerkin schemes with application to phase-field methods”
Abstract
Many mathematical CFD models involve transport of conserved quantities that must lie in a certain range to be physically meaningful. The solution u of a scalar conservation law is said to satisfy a maximum principle (MP) if global bounds u_min and u_max exist such that u_min <= u(t,x) <= u_max. To enforce such inequality constraints for DG solutions at least for element averages, the numerical fluxes must be defined and constrained in an appropriate manner. We introduce a general framework for calculating fluxes that produce non-oscillatory DG approximations and preserve the MP for element averages even if the exact solution of the PDE violates them due to modeling errors or perturbed data. The proposed methodology is based on a combination of flux and slope limiting: The flux limiter constrains changes of element averages so as to prevent violations of global bounds. The subsequent slope limiting step adjusts the higher order solution parts to impose local bounds on pointwise values of the high-order DG solution. Since manipulations of the target flux may introduce a consistency error, it is essential to guarantee that physically admissible fluxes remain unchanged. The novel fractional step flux limiting approach is iterative while in each iteration, the MP property is guaranteed and the consistency error is reduced. Practical applicability is demonstrated by numerical studies for the advection equation (hyperbolic, linear) and the Cahn-Hilliard equation (parabolic, nonlinear). The flux limiter (similar to slope limiter) is a local/parallelizable postprocessing procedure that can be applied to various types of DG discretizations of a wide range of scalar conservation laws.