Local Discontinuous Galerkin methods for fractional diffusion equations

W. H. Deng; J. S. Hesthaven

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 6, page 1845-1864
  • ISSN: 0764-583X

Abstract

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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 β ∈[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(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion (β = 2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.

How to cite

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Deng, W. H., and Hesthaven, J. S.. "Local Discontinuous Galerkin methods for fractional diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.6 (2013): 1845-1864. <http://eudml.org/doc/273249>.

@article{Deng2013,
abstract = {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 β ∈[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 &#x1d4aa;(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion (β = 2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.},
author = {Deng, W. H., Hesthaven, J. S.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {fractional derivatives; local discontinuous Galerkin methods; stability; convergence; error estimates},
language = {eng},
number = {6},
pages = {1845-1864},
publisher = {EDP-Sciences},
title = {Local Discontinuous Galerkin methods for fractional diffusion equations},
url = {http://eudml.org/doc/273249},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Deng, W. H.
AU - Hesthaven, J. S.
TI - Local Discontinuous Galerkin methods for fractional diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 6
SP - 1845
EP - 1864
AB - 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 β ∈[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 &#x1d4aa;(hk + 1) uniformly across the continuous range between pure advection (β = 1) and pure diffusion (β = 2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.
LA - eng
KW - fractional derivatives; local discontinuous Galerkin methods; stability; convergence; error estimates
UR - http://eudml.org/doc/273249
ER -

References

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