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Local Discontinuous Galerkin methods for fractional diffusion equations

W. H. Deng;  J. S. Hesthaven — 2013

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by  ∈[1, 2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence 𝒪( ) uniformly across the continuous range between pure advection ( = 1) and pure...

Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method

A. Klöckner;  T. Warburton;  J. S. Hesthaven — 2011

Mathematical Modelling of Natural Phenomena

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the...

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