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What is a limit of structure-preserving numerical methods for compressible flows?

Maria Lukacova-Medvidova, Bangwei She, Yuhuan Yuan

TL;DR

The paper addresses the fundamental question of what limits should be expected from structure-preserving numerical methods for inviscid multidimensional compressible flows. It develops a framework based on dissipative solutions and $K$-convergence, linking numerical defects—namely the Reynolds-stress defect and the energy defect—to convergence behavior. Through numerical experiments, it demonstrates how structure-preserving schemes can converge to dissipative solutions and how defects quantify the approach. This work provides a theoretical and computational bridge between rigorous convergence analysis and turbulent-flow numerics, guiding the design of physically faithful discretizations.

Abstract

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.

What is a limit of structure-preserving numerical methods for compressible flows?

TL;DR

The paper addresses the fundamental question of what limits should be expected from structure-preserving numerical methods for inviscid multidimensional compressible flows. It develops a framework based on dissipative solutions and -convergence, linking numerical defects—namely the Reynolds-stress defect and the energy defect—to convergence behavior. Through numerical experiments, it demonstrates how structure-preserving schemes can converge to dissipative solutions and how defects quantify the approach. This work provides a theoretical and computational bridge between rigorous convergence analysis and turbulent-flow numerics, guiding the design of physically faithful discretizations.

Abstract

We present an overview of recent developments on the convergence analysis of numerical methods for inviscid multidimensional compressible flows that preserve underlying physical structures. We introduce the concept of generalized solutions, the so-called dissipative solutions, and explain their relationship to other commonly used solution concepts. In numerical experiments we apply K-convergence of numerical solutions and approximate turbulent solutions together with the Reynolds stress defect and the energy defect.
Paper Structure (10 sections, 2 theorems, 3 equations, 2 figures, 1 table)

This paper contains 10 sections, 2 theorems, 3 equations, 2 figures, 1 table.

Table of Contents

  1. Section Heading
  2. Section Heading
  3. Subsection Heading
  4. Subsubsection Heading
  5. Paragraph Heading
  6. Subparagraph Heading
  7. Section Heading
  8. Subsection Heading
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  10. Paragraph Heading

Key Result

theorem 1

Theorem text goes here.

Figures (2)

  • Figure 1: If the width of the figure is less than 7.8 cm use the sidecapion command to flush the caption on the left side of the page. If the figure is positioned at the top of the page, align the sidecaption with the top of the figure -- to achieve this you simply need to use the optional argument [t] with the sidecaption command
  • Figure 2: If the width of the figure is less than 7.8 cm use the sidecapion command to flush the caption on the left side of the page. If the figure is positioned at the top of the page, align the sidecaption with the top of the figure -- to achieve this you simply need to use the optional argument [t] with the sidecaption command

Theorems & Definitions (6)

  • theorem 1
  • definition 1
  • proof
  • theorem 2
  • definition 2
  • proof