scholarly journals Using Lie group integrators to solve two and higher dimensional variational problems with symmetry

2019 ◽  
Vol 6 (2) ◽  
pp. 485-511
Author(s):  
Michele Zadra ◽  
◽  
Elizabeth L. Mansfield
Keyword(s):  
Lie Group ◽  
Variational Problems ◽  
Lie Group Integrators ◽  
Higher Dimensional

scholarly journals Construction and Characterization of Representations of SU(7) for GUT Model Builders

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1044
Author(s):  
Daniel Jones ◽  
Jeffery A. Secrest
Keyword(s):  
Gauge Symmetry ◽  
Symmetry Group ◽  
Lie Group ◽  
Cartan Subalgebra ◽  
Natural Extension ◽  
Grand Unification ◽  
Raising And Lowering Operators ◽  
Higher Dimensional ◽  
Lowering Operators

The natural extension to the SU(5) Georgi-Glashow grand unification model is to enlarge the gauge symmetry group. In this work, the SU(7) symmetry group is examined. The Cartan subalgebra is determined along with their commutation relations. The associated roots and weights of the SU(7) algebra are derived and discussed. The raising and lowering operators are explicitly constructed and presented. Higher dimensional representations are developed by graphical as well as tensorial methods. Applications of the SU(7) Lie group to supersymmetric grand unification as well as applications are discussed.

Explicit Lie group integrators

1991 ◽  
Author(s):  
Étienne Forest ◽  
James Murphy ◽  
Michael F. Reusch
Keyword(s):  
Lie Group ◽  
Lie Group Integrators

scholarly journals Variable step size commutator free Lie group integrators

2019 ◽  
Vol 82 (4) ◽  
pp. 1359-1376
Author(s):  
Charles Curry ◽  
Brynjulf Owren
Keyword(s):  
Lie Group ◽  
Variable Step Size ◽  
Step Size ◽  
Lie Group Integrators ◽  
Variable Step

scholarly journals Lie Group integrators for mechanical systems

2021 ◽  
pp. 1-38
Author(s):  
Elena Celledoni ◽  
Ergys Çokaj ◽  
Andrea Leone ◽  
Davide Murari ◽  
Brynjulf Owren
Keyword(s):  
Lie Group ◽  
Mechanical Systems ◽  
Lie Group Integrators

scholarly journals Convergence of Lie group integrators

2019 ◽  
Vol 144 (2) ◽  
pp. 357-373 ◽  
Author(s):  
Charles Curry ◽  
Alexander Schmeding
Keyword(s):  
Lie Group ◽  
Lie Group Integrators

scholarly journals Topics in structure-preserving discretization

Acta Numerica ◽  
2011 ◽  
Vol 20 ◽  
pp. 1-119 ◽  
Author(s):  
Snorre H. Christiansen ◽  
Hans Z. Munthe-Kaas ◽  
Brynjulf Owren
Keyword(s):  
Partial Differential Equations ◽  
Finite Elements ◽  
Differential Equations ◽  
Lie Group ◽  
Variable Order ◽  
Partial Differential ◽  
Structure Preserving ◽  
Hodge Laplacian ◽  
Lie Group Integrators ◽  
Inverse Systems

In the last few decades the concepts of structure-preserving discretization, geometric integration and compatible discretizations have emerged as subfields in the numerical approximation of ordinary and partial differential equations. The article discusses certain selected topics within these areas; discretization techniques both in space and time are considered. Lie group integrators are discussed with particular focus on the application to partial differential equations, followed by a discussion of how time integrators can be designed to preserve first integrals in the differential equation using discrete gradients and discrete variational derivatives.Lie group integrators depend crucially on fast and structure-preserving algorithms for computing matrix exponentials. Preservation of domain symmetries is of particular interest in the application of Lie group integrators to PDEs. The equivariance of linear operators and Fourier transforms on non-commutative groups is used to construct fast structure-preserving algorithms for computing exponentials. The theory of Weyl groups is employed in the construction of high-order spectral element discretizations, based on multivariate Chebyshev polynomials on triangles, simplexes and simplicial complexes.The theory of mixed finite elements is developed in terms of special inverse systems of complexes of differential forms, where the inclusion of cells corresponds to pullback of forms. The theory covers, for instance, composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge–Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

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