Volume 23, Issue 3
An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation

Nianyu Yi & Hailiang Liu

Commun. Comput. Phys., 23 (2018), pp. 747-772.

Published online: 2018-03

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  • Abstract

We design and numerically validate a local discontinuous Galerkin (LDG) method to compute solutions to the initial value problem for a nonlinear variational wave equation originally proposed to model liquid crystals. For the semi-discrete LDG formulation with a class of alternating numerical fluxes, the energy conserving property is verified. A dissipative scheme is also introduced by locally imposing some numerical "damping" in the scheme so as to suppress some numerical oscillations near solution singularities. Extensive numerical experiments are presented to validate and illustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ is numerically obtained when using alternating numerical fluxes. When using the central numerical flux, only sub-optimal convergence is observed for polynomials of odd degree. Numerical simulations with long time integration indicate that the energy conserving property is crucial for accurately capturing the underlying wave shapes.

  • Keywords

Discontinuous Galerkin method, variational wave equation, energy conservation.

  • AMS Subject Headings

65M60, 65M12, 35Q35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-23-747, author = {}, title = {An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation}, journal = {Communications in Computational Physics}, year = {2018}, volume = {23}, number = {3}, pages = {747--772}, abstract = {

We design and numerically validate a local discontinuous Galerkin (LDG) method to compute solutions to the initial value problem for a nonlinear variational wave equation originally proposed to model liquid crystals. For the semi-discrete LDG formulation with a class of alternating numerical fluxes, the energy conserving property is verified. A dissipative scheme is also introduced by locally imposing some numerical "damping" in the scheme so as to suppress some numerical oscillations near solution singularities. Extensive numerical experiments are presented to validate and illustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ is numerically obtained when using alternating numerical fluxes. When using the central numerical flux, only sub-optimal convergence is observed for polynomials of odd degree. Numerical simulations with long time integration indicate that the energy conserving property is crucial for accurately capturing the underlying wave shapes.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0189}, url = {http://global-sci.org/intro/article_detail/cicp/10546.html} }
TY - JOUR T1 - An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation JO - Communications in Computational Physics VL - 3 SP - 747 EP - 772 PY - 2018 DA - 2018/03 SN - 23 DO - http://doi.org/10.4208/cicp.OA-2016-0189 UR - https://global-sci.org/intro/article_detail/cicp/10546.html KW - Discontinuous Galerkin method, variational wave equation, energy conservation. AB -

We design and numerically validate a local discontinuous Galerkin (LDG) method to compute solutions to the initial value problem for a nonlinear variational wave equation originally proposed to model liquid crystals. For the semi-discrete LDG formulation with a class of alternating numerical fluxes, the energy conserving property is verified. A dissipative scheme is also introduced by locally imposing some numerical "damping" in the scheme so as to suppress some numerical oscillations near solution singularities. Extensive numerical experiments are presented to validate and illustrate the effectiveness of the numerical methods. Optimal convergence in $L^2$ is numerically obtained when using alternating numerical fluxes. When using the central numerical flux, only sub-optimal convergence is observed for polynomials of odd degree. Numerical simulations with long time integration indicate that the energy conserving property is crucial for accurately capturing the underlying wave shapes.

Nianyu Yi & Hailiang Liu. (2020). An Energy Conserving Local Discontinuous Galerkin Method for a Nonlinear Variational Wave Equation. Communications in Computational Physics. 23 (3). 747-772. doi:10.4208/cicp.OA-2016-0189
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