Core Concepts
Implicit boundary conditions in PDE discretizations lead to spurious modes, identified through quality tests.
Abstract
The article discusses the issue of spurious modes in PDE discretizations due to implicit boundary conditions. It introduces the concept of additional implicit constraints and quality measures to identify and eliminate spurious modes. The study focuses on linear problems and proposes tests based on violations of derivatives of boundary conditions and Grassmann distance between subspaces. The effectiveness of these tests is demonstrated through numerical experiments on various examples.
- Introduction:
- Spurious modes in PDE discretizations.
- Importance of identifying and removing spurious modes.
- Motivation:
- Boundary conditions as algebraic constraints.
- Interaction of algebraic constraints and differential equations.
- Eliminating and Identifying Spurious Modes:
- Application of Algorithm 1 to enforce implicit constraints.
- Development of quality measures: derivative test and angle criterion.
- Tests for Identifying Spurious Modes:
- Boundary derivative criterion to measure violation of implicit constraints.
- Angle criterion using Grassmann distance for eigenvalue problems.
- Generalized Eigenvalue Problems:
- Extension of angle criterion to handle generalized eigenvalue problems.
- Combining Quality Criteria with Implicit Constraints:
- Application of additional implicit constraints with quality measures.
Stats
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Quotes
"In most discretization schemes, these additional implicit boundary conditions are violated."
"The Grassmann distance criterion clearly identifies the portion of the spectrum which is accurately captured versus the spurious parts."