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Numerical Boundary Control of Multi-Dimensional Hyperbolic Equations

Michael Herty, Kai Hinzmann, Siegfried Müller, Ferdinand Thein

Abstract

Existing theoretical stabilization results for linear, hyperbolic multi-dimensional problems are extended to the discretized multi-dimensional problems. In contrast to existing theoretical and numerical analysis in the spatially one-dimensional case the effect of the numerical dissipation is analyzed and explicitly quantified. Further, using dimensional splitting, the numerical analysis is extended to the multi-dimensional case. The findings are confirmed by numerical simulations for low-order and high-order DG schemes both in the one-dimensional and two-dimensional case.

Numerical Boundary Control of Multi-Dimensional Hyperbolic Equations

Abstract

Existing theoretical stabilization results for linear, hyperbolic multi-dimensional problems are extended to the discretized multi-dimensional problems. In contrast to existing theoretical and numerical analysis in the spatially one-dimensional case the effect of the numerical dissipation is analyzed and explicitly quantified. Further, using dimensional splitting, the numerical analysis is extended to the multi-dimensional case. The findings are confirmed by numerical simulations for low-order and high-order DG schemes both in the one-dimensional and two-dimensional case.

Paper Structure

This paper contains 11 sections, 4 theorems, 104 equations, 5 figures, 2 tables.

Table of Contents

  1. Description of the Problem
  2. Feedback Control Problem with Viscosity
  3. Numerical analysis of the discretized problem
  4. Proof of Theorem \ref{['thm:decay_discr_Lyapunov_theorem-multid']}
  5. Setting and one--dimensional analysis
  6. Decay rate of the discrete Lyapunov function
  7. Multi--dimensional analysis
  8. Numerical results
  9. Investigations in the 1D case
  10. Multi--d stabilization using higher order schemes
  11. Outlook

Key Result

Theorem 1.3

Let $\mathbf{w}\in C^1(0,T; H^s(\Omega))^m$ for $s\geq 1+\frac{d}{2}$ be a solution to IBVP eq:IBVP. A Lyapunov function is given by where for functions $\mu_i(\mathbf{x})\in H^s(\Omega)$ for $i\in\{1,\dots,m\}$ that satisfy for some $C_L\in\mathbb{R}_{>0}$, where Here, $\mathcal{D}$ is assumed to be a constant matrix which satisfies Then, for every $\mathbf{u}(t,\mathbf{x}) = (u_1(t,\mathbf{

Figures (5)

  • Figure 1: Decay of the discrete Lyapunov function $\mathcal{L}^n$ for schemes with $q=1$ (top), $q=\lambda a=0.5$ (middle) and $q=(\lambda a)^2=0.25$ (bottom) and fixed $\operatorname*{CFL} = 0.5$ compared to the discretized decay rate without diffusion $\operatorname*{exp}(-C_L n\Delta t)\mathcal{L}^0$ and the discrete decay rates of Theorem \ref{['decay_discr_Lyapunov_theorem']}. Results are shown for $10$ equidistantly distributed time steps in $[0,T]$.
  • Figure 2: Plots of the discrete control $\left(u^n\right)_{n=0,\dots,N}$ and the residual $\left(\mathcal{R}_n\right)_{n=0,\dots,N}$ for different values of $q$ with $\operatorname*{CFL}=0.5$ and $\Delta x = 0.001$.
  • Figure 3: Decay of the approximated Lyapunov function $\hat{L}(t)$ for a simulation with $P=1$ (left) and $P=2$ (right), $\operatorname*{CFL} = 0.7$, $\overline{L}=6$ and $c_{thresh} = 0.1$ compared to the exact decay rate $L(0)\operatorname*{exp}(-C_L t)$ in Theorem \ref{['Hauptresultat']}.
  • Figure 4: Convergence of the approximated Lyapunov function $\hat{L}(t)$ for different maximum refinement levels for simulations with $P=1$, $P=2$ and $P=3$ with $\operatorname*{CFL} = 0.7$ with grid adaptation ($c_{thresh} = 0.1$) to the exact one $L(0)\operatorname*{exp}(-C_L t)$ at $T=3$.
  • Figure : $t=0.0$

Theorems & Definitions (12)

  • Remark 1.1
  • Definition 1.2: Exponentially stable
  • Theorem 1.3: Exponential stability
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Theorem 1.7: Decay of the discrete Lyapunov function in multi--dimension
  • Theorem 2.1
  • proof
  • ...and 2 more