Mimicking of Non-conservative Systems

Abstract

The mimetic spectral element method is a relatively young method in numerical solutions of partial differential equations and actions that describe physical systems. Its advantage is that it takes the geometrical structure of the problem into account which guarantees consistency of the numerical scheme and the conservation of relevant quantities. This prohibits the solution to diverge into non-physical states since the discretization mimics the physical behaviour of the problem. With relation to physics the mimetic method makes use of the stationary action principle which contains all information about the system and reduces the continuous variables to discrete ones that become ready for computation. The stationary action principle requires a formulation of the Lagrangian such that it implies Newton’s laws of motion or equivalent physical laws. However these Lagrangians are only easily formulated for monogenic systems. This work investigates the class of (discrete) non-monogenic physical systems. The goal is to present a general method to numerically mimic non-conservative systems such that energy (loss) is exact. This is done by modifying a newly introduced systematic method that transforms the non-conservative problem to a system with a doubled degrees of freedom that makes it a conservative or monogenic one. In this modification process new degrees of freedom are introduced which are allowed to have arbitrary variations at the initial and final states of the new action. The system can than be mimicked with the spectral element method where algebraic dual polynomials are used as both approximating polynomials and as test functions that serve as arbitrary variations. Furthermore a gauge transformation is introduced by the modified action such that it contains the initial and final states of the system to make the solution unique.
The introduced method is generally applicable to Lagrangian mechanics of discrete systems with holonomic constraints. The method is applied to the damped harmonic oscillator (DHO) as a test case. The computational results are clearly accurate and converging. However they deviate with one order from the h-refinement spectral element theory of Gauss-Lobatto-Legendre interpolation basis polynomials. Theoretical analysis of the equations of the mimicked (single and doubled) DHO show that energy (loss) is exact. Nevertheless, the computations show an oscillation in energy that remains closely around the conserved value. This oscillation in energy in time was observed in earlier works that applied the mimetic spectral element method to conservative systems. The reason behind this behavior of energy remains a topic for further research. Furthermore, in a follow up research the method can be extended to the Lagrangian field theory and tested with simplified non-conservative fluid problems to verify it for fields.