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New Low-Dissipation Central-Upwind Schemes. Part II

Shaoshuai Chu, Alexander Kurganov, Ruixiao Xin

TL;DR

The paper addresses spurious oscillations observed in low-dissipation central-upwind schemes when solving hyperbolic conservation laws, particularly near boundaries. It introduces a simple, systematic modification of the projection step by adding a free shift of the projection point, yielding a NEW LDCU framework that preserves high resolution while suppressing oscillations. The NEW scheme is developed for both 1-D and 2-D Euler equations, with fully discrete and semi-discrete formulations that incorporate built-in anti-diffusion terms, and it is demonstrated to produce non-oscillatory solutions with comparable or improved accuracy across a suite of test problems. This approach enhances robustness and accuracy for high-resolution finite-volume simulations of gas-dynamic flows and related hyperbolic systems, especially near domain boundaries.

Abstract

The low-dissipation central-upwind (LDCU) schemes have been recently introduced in [A. Kurganov and R. Xin, J. Sci. Comput., 96 (2023), Paper No. 56] as a modification of the central-upwind (CU) schemes from [{\sc A. Kurganov and C. T. Lin, Commun. Comput. Phys., 2 (2007), pp. 141-163}]. The LDCU schemes achieve much higher resolution of contact waves and many (two-dimensional) structures resulting from complicated wave interaction. However, the LDCU schemes sometimes produce more oscillatory results compared with the CU schemes, especially near the computational domain boundaries. In this paper, we propose a very simple -- yet systematic -- modification of the LDCU schemes, which completely eliminates the aforementioned oscillations almost without affecting the quality of the computed solution.

New Low-Dissipation Central-Upwind Schemes. Part II

TL;DR

The paper addresses spurious oscillations observed in low-dissipation central-upwind schemes when solving hyperbolic conservation laws, particularly near boundaries. It introduces a simple, systematic modification of the projection step by adding a free shift of the projection point, yielding a NEW LDCU framework that preserves high resolution while suppressing oscillations. The NEW scheme is developed for both 1-D and 2-D Euler equations, with fully discrete and semi-discrete formulations that incorporate built-in anti-diffusion terms, and it is demonstrated to produce non-oscillatory solutions with comparable or improved accuracy across a suite of test problems. This approach enhances robustness and accuracy for high-resolution finite-volume simulations of gas-dynamic flows and related hyperbolic systems, especially near domain boundaries.

Abstract

The low-dissipation central-upwind (LDCU) schemes have been recently introduced in [A. Kurganov and R. Xin, J. Sci. Comput., 96 (2023), Paper No. 56] as a modification of the central-upwind (CU) schemes from [{\sc A. Kurganov and C. T. Lin, Commun. Comput. Phys., 2 (2007), pp. 141-163}]. The LDCU schemes achieve much higher resolution of contact waves and many (two-dimensional) structures resulting from complicated wave interaction. However, the LDCU schemes sometimes produce more oscillatory results compared with the CU schemes, especially near the computational domain boundaries. In this paper, we propose a very simple -- yet systematic -- modification of the LDCU schemes, which completely eliminates the aforementioned oscillations almost without affecting the quality of the computed solution.
Paper Structure (13 sections, 41 equations, 6 figures)

This paper contains 13 sections, 41 equations, 6 figures.

Table of Contents

  1. Introduction
  2. New LDCU Schemes
  3. Modification of the Projection Step
  4. New LDCU Scheme for the 1-D Euler Equations of Gas Dynamics
  5. Fully Discrete Scheme
  6. Semi-Discrete Scheme
  7. New LDCU Scheme for the 2-D Euler Equations of Gas Dynamics
  8. "Built-in" Anti-Diffusion
  9. Numerical Examples
  10. Example 5---Implosion Problem
  11. Funding.
  12. Conflicts of interest.
  13. Data and software availability.

Figures (6)

  • Figure 2.1: The new projection step.
  • Figure 3.1: Example 1: Density $\rho$ computed by the OLD and NEW schemes (left) and zoom at $x\in[-5,-2]$ (right).
  • Figure 3.2: Example 2: Density $\rho$ computed by the OLD and NEW schemes on two uniform meshes (left) and zoom at $x\in[0.7,0.9]$ and $x\in[0.8,0.801]$ (right).
  • Figure 3.3: Example 3: Density ($\rho$) computed by the OLD (left) and NEW (right) schemes.
  • Figure 3.4: Example 4: Density ($\rho$) computed by the OLD (left) and NEW (right) schemes.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 2.2