Global Bounds for the Error in Solutions of Linear Hyperbolic Systems due to Inaccurate Boundary Geometry

TL;DR

The paper analyzes how inaccuracies in boundary geometry affect solutions to linear hyperbolic systems by deriving global $L^2$ energy bounds for both the correct and erroneous domains. It shows that the dominant error sources are Jacobian and metric-term discrepancies, with boundary-data evaluation errors being secondary, and that in two dimensions the leading error terms are proportional to boundary curve location errors and their derivatives. The authors formulate an error equation for $\mathbf e = \mathbf v - \mathbf q$ on a reference domain and obtain long-time bounds that depend on geometry perturbations through quantities like $|\Delta\Gamma|$ and $|\Delta\Gamma'|$, highlighting the role of boundary dissipation in controlling growth. The results have practical implications for high-order mesh generation and optimization of boundary approximations, indicating that boundary geometry must converge at the solver’s rate to achieve expected accuracy and that accurate normals and Jacobians are essential for controlling error propagation in hyperbolic problems.

Abstract

We derive global estimates for the error in solutions of linear hyperbolic systems due to inaccurate boundary geometry. We show that the error is bounded by data and bounded in time when the solutions in the true and approximate domains are bounded. We show that boundary data evaluation errors due to the incorrect locations of the boundaries are secondary effects, whereas the primary errors are from the Jacobian and metric terms. In two space dimensions, specifically, we show that to lowest order the errors are proportional to the errors in the boundary curves and their derivatives. The results illustrate the importance of accurately approximating boundaries, and they should be helpful for high-order mesh generation and the design of optimization algorithms for boundary approximations.

Figures (11)

• Figure 1: Diagram of the correct, $\Omega$, erroneous, $\Omega_e$, and reference, $\mathcal{D}$, domains
• Figure 2: Correct domain (solid lines) and Erroneous domain (dashed line) boundary curves in two space dimensions used to isolate the boundary errors to one boundary
• Figure 3: Time history of the error for two values of the left boundary location error, $\delta$, and for two values of $\gamma$ for the one dimensional scalar problem, \ref{['eq:1DScalarProblem']}
• Figure 4: Variation of the maximum errors as a function of $\delta$ for the large time solution of the one dimensional scalar problem, \ref{['eq:1DScalarProblem']}
• Figure 5: Correct and erroneous domains in two space dimensions with linear and quadratic perturbations. On the left, the approximate curve matches at least one end point. On the right is the minimax approximation

Theorems & Definitions (10)

• Remark 1
• Remark 2
• Example 1
• Remark 3
• Remark 4
• Remark 5
• Theorem 1
• proof
• Remark 6
• Remark 7