Energy-conserving spectral element schemes based on Lagrangian dynamics

Investigating discrete energy conservation

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Abstract

Physical systems in the continuous domain are often solved using computer-aided software because of their complexity. Preserving the physical quantities from the continuous domain in the discrete domain is therefore of utmost importance. There is however a broad range of techniques that can accomplish the translation between the continuous and discrete domain, e.g. finite difference, volume and element techniques, fourth order Runge-Kutta or Störmer-Verlet to name a few. Accompanied with the aforementioned come strengths and weaknesses but have the common thread to try maintaining the physical behaviour of the continuous system closely. The mimetic spectral element technique is used to develop an energy-conserving spectral element scheme through a Lagrangian formulation. This new formulation of the mimetic spectral element technique allows for solving time-dependent problems and the simple harmonic oscillator serves as the sample problem in this thesis. The solution has been derived from Lagrangian mechanics in a variety of ways. Discrete Lagrangian formulations have been investigated at rst and their respective equations of motions have been tested against the exact solution of the simple harmonic oscillator. This method achieved marching in time and no damping of the solution, yet energy was only bounded and not exactly conserved. The mimetic spectral element formulation of the Lagrangian formulation showed diculties when using variational analysis, i.e. boundary treatment in the future. Arbitrary domain mapping was among the possible solutions, but this formulation was found to be unreliable and unsuccessful. It was found that a more robust formulation, i.e. the spectral marching method, was most suitable. Throughout this thesis the focus was put on the conservation of energy using a Lagrangian formulation. Except the spectral scheme using arbitrary domain mapping, all schemes kept the energy of the system bounded, but no energy was conserved up to machine precision. Using arbitrary domain mapping, the energy seemed to grow over time.

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