A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

Abigail Wacher

Open Mathematics (2013)

  • Volume: 11, Issue: 4, page 642-663
  • ISSN: 2391-5455

Abstract

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We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.

How to cite

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Abigail Wacher. "A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations." Open Mathematics 11.4 (2013): 642-663. <http://eudml.org/doc/269308>.

@article{AbigailWacher2013,
abstract = {We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.},
author = {Abigail Wacher},
journal = {Open Mathematics},
keywords = {Moving meshes; Weighted moving finite elements; Moving mesh partial differential equations; Numerical solutions of partial differential equations; Porous medium equation; Waiting-time solutions; Viscous Burgers’ equation; moving meshes; weighted moving finite elements; moving mesh partial differential equations; porous medium equation; waiting-time solutions; viscous Burgers' equation; error bounds; numerical experiments; travelling wave solution; Barenblatt selfsimilar analytical solution},
language = {eng},
number = {4},
pages = {642-663},
title = {A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations},
url = {http://eudml.org/doc/269308},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Abigail Wacher
TI - A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations
JO - Open Mathematics
PY - 2013
VL - 11
IS - 4
SP - 642
EP - 663
AB - We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both in one dimension. Simulations are made for the two methods for: a) a travelling wave solution for the viscous Burgers’ equation, b) the Barenblatt selfsimilar analytical solution of the porous medium equation, and c) a waiting-time solution for the porous medium equation. Simulations are carried out for varying mesh sizes, and the numerical solutions are compared by computing errors in two ways. In the case of an analytic solution being available, the errors in the numerical solutions are computed directly from the analytic solution. In the case of no availability of an analytic solution, an approximation to the error is computed using a very fine mesh numerical solution as the reference solution.
LA - eng
KW - Moving meshes; Weighted moving finite elements; Moving mesh partial differential equations; Numerical solutions of partial differential equations; Porous medium equation; Waiting-time solutions; Viscous Burgers’ equation; moving meshes; weighted moving finite elements; moving mesh partial differential equations; porous medium equation; waiting-time solutions; viscous Burgers' equation; error bounds; numerical experiments; travelling wave solution; Barenblatt selfsimilar analytical solution
UR - http://eudml.org/doc/269308
ER -

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