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Volume 14, Issue 1
High-Order Energy Stable Numerical Schemes for a Nonlinear Variational Wave Equation Modeling Nematic Liquid Crystals in Two Dimensions

P. Aursand & U. Koley

Int. J. Numer. Anal. Mod., 14 (2017), pp. 20-47.

Published online: 2016-01

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  • Abstract

We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic property of solutions to this model. For the two-dimensional case, we design numerical schemes based on the discontinuous Galerkin framework that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments are performed verifying the scheme's energy stability, order of convergence and computational efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge to different solutions.

  • Keywords

Nonlinear variational wave equation, energy preserving scheme, energy stable scheme, discontinuous Galerkin method, higher order scheme.

  • AMS Subject Headings

Primary 65M99, Secondary 65M60, 35L60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-20, author = {P. Aursand and U. Koley}, title = {High-Order Energy Stable Numerical Schemes for a Nonlinear Variational Wave Equation Modeling Nematic Liquid Crystals in Two Dimensions}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {14}, number = {1}, pages = {20--47}, abstract = {

We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic property of solutions to this model. For the two-dimensional case, we design numerical schemes based on the discontinuous Galerkin framework that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments are performed verifying the scheme's energy stability, order of convergence and computational efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge to different solutions.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/408.html} }
TY - JOUR T1 - High-Order Energy Stable Numerical Schemes for a Nonlinear Variational Wave Equation Modeling Nematic Liquid Crystals in Two Dimensions AU - P. Aursand & U. Koley JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 20 EP - 47 PY - 2016 DA - 2016/01 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/408.html KW - Nonlinear variational wave equation, energy preserving scheme, energy stable scheme, discontinuous Galerkin method, higher order scheme. AB -

We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic property of solutions to this model. For the two-dimensional case, we design numerical schemes based on the discontinuous Galerkin framework that either conserve or dissipate a discrete version of the energy. Extensive numerical experiments are performed verifying the scheme's energy stability, order of convergence and computational efficiency. The numerical solutions are compared to those of a simpler first-order Hamiltonian scheme. We provide numerical evidence that solutions of the 2D variational wave equation loose regularity in finite time. After that occurs, dissipative and conservative schemes appear to converge to different solutions.

P. Aursand and U. Koley. (2016). High-Order Energy Stable Numerical Schemes for a Nonlinear Variational Wave Equation Modeling Nematic Liquid Crystals in Two Dimensions. International Journal of Numerical Analysis and Modeling. 14 (1). 20-47. doi:
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