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Shock-capturing with natural high frequency oscillations

Y. C. Zhou, Yun Gu, G. W. Wei

TL;DR

This paper tackles the challenge of high-accuracy, low-dissipation shock-capturing for compressible Euler equations that include natural high-frequency components. It introduces the CFOR scheme built on discrete singular convolution (DSC), which separates high-frequency spatial discretization (high-pass) from post-processing (low-pass) guided by a wavelet-based sensor to suppress spurious oscillations. The authors demonstrate strong performance on a 2D isentropic vortex and on shock-entropy wave interactions, achieving spectral-like accuracy, long-time stability, and robust shock resolution with minimal dissipation. Overall, CFOR offers a controllable-accuracy alternative to traditional high-order schemes for problems featuring shocks and high-frequency content, with potential for adaptive optimization and broader applicability to hyperbolic conservation laws.

Abstract

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock-capturing under the influence of natural high-frequency oscillations. The conjugate low-pass and high-pass filters are constructed based on the principle of the discrete singular convolution. Two Euler systems, the advection of an isentropy vortex flow and the interaction of shock-entropy wave are considered to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock-entropy wave interaction. Numerical experiments indicate that the proposed scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws.

Shock-capturing with natural high frequency oscillations

TL;DR

This paper tackles the challenge of high-accuracy, low-dissipation shock-capturing for compressible Euler equations that include natural high-frequency components. It introduces the CFOR scheme built on discrete singular convolution (DSC), which separates high-frequency spatial discretization (high-pass) from post-processing (low-pass) guided by a wavelet-based sensor to suppress spurious oscillations. The authors demonstrate strong performance on a 2D isentropic vortex and on shock-entropy wave interactions, achieving spectral-like accuracy, long-time stability, and robust shock resolution with minimal dissipation. Overall, CFOR offers a controllable-accuracy alternative to traditional high-order schemes for problems featuring shocks and high-frequency content, with potential for adaptive optimization and broader applicability to hyperbolic conservation laws.

Abstract

This paper explores the potential of a newly developed conjugate filter oscillation reduction (CFOR) scheme for shock-capturing under the influence of natural high-frequency oscillations. The conjugate low-pass and high-pass filters are constructed based on the principle of the discrete singular convolution. Two Euler systems, the advection of an isentropy vortex flow and the interaction of shock-entropy wave are considered to demonstrate the utility of the CFOR scheme. Computational accuracy and order of approximation are examined and compared with the literature. Some of the best numerical results are obtained for the shock-entropy wave interaction. Numerical experiments indicate that the proposed scheme is stable, conservative and reliable for the numerical simulation of hyperbolic conservation laws.

Paper Structure

This paper contains 8 sections, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The DSC algorithm and CFOR scheme
  3. DSC filters
  4. The CFOR scheme
  5. Results and discussions
  6. Isentropic vortex
  7. Interaction of shock and high-frequency entropy wave
  8. Conclusion

Figures (3)

  • Figure 1: Frequency responses of the conjugate DSC filters (in the unit of $\pi/\Delta$), The maximum amplitude of the filters is normalized to the unit.
  • Figure 2: Density profiles in horizontal cutting at four times. Solid line is the exact profile and the circle dots denote the numerical results.
  • Figure 3: Shock entropy wave interaction.