Scattering-Passive Structure-Preserving Finite Element Method for the Boundary Controlled Transport Equation with a Moving Mesh

TL;DR

The paper develops a structure-preserving finite element framework for transport equations in 1D and 2D that preserves the scattering-energy balance $\dot{H}(t)=\dfrac{1}{2}\left(u(t)^2-y(t)^2\right)$ via a discrete Hamiltonian $H(t)=\dfrac{1}{2}\int_a^b \mathcal{H}(\zeta)x(\zeta,t)^2 d\zeta$. It provides FEM discretizations that mimic this energy structure in both dimensions, with explicit DAEs in 1D and a flux-form discretization in 2D, and shows energy preservation holds when $\operatorname{div}(\bold{c})=0$ (with a dissipative term if not). A moving-mesh extension for 1D is introduced, yielding time-varying matrices $E(t)$ and $Q(t)$ and a transformed energy balance that remains consistent with the continuous model; this extension can reduce Gibbs phenomena and improve accuracy. Numerical simulations illustrate the moving mesh outperforming a fixed mesh in terms of accuracy and oscillation reduction, and the framework is positioned for integration with control and predictive-scheme design, underpinned by a natural Lyapunov-like energy function for stability analysis.

Abstract

A structure-preserving Finite Element Method (FEM) for the transport equation in one- and two-dimensional domains is presented. This Distributed Parameter System (DPS) has non-collocated boundary control and observation, and reveals a scattering-energy preserving structure. We show that the discretized model preserves the aforementioned structure from the original infinite-dimensional system. Moreover, we analyse the case of moving meshes for the one-dimensional case. The moving mesh requires less states than the fixed one to produce solutions with a comparable accuracy, and it can also reduce the overshoot and oscillations of Gibbs phenomenon produced when using the FEM. Numerical simulations are provided for the case of a one-dimensional transport equation with fixed and moving meshes.

Figures (10)

• Figure 1: Spatial domain and boundary partitions.
• Figure 2: Basis functions $\Phi(\zeta)$ for the fixed mesh.
• Figure 3: Initial condition of the basis functions $\Phi(\zeta,0)$ for the moving mesh.
• Figure 4: Node trajectories. Dashed lines represent the fixed case and solid ones the moving case.
• Figure 5: Output of the system $y(t) = x(a,t)$.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5