A simple, high-order and compact WENO limiter based on control volume for spectral volume method

Abstract

The spectral volume(SV) method constructs a high-order polynomial for SV based on the average value of control volume(CV), but for discontinuous problems, a limiter is required to mitigate oscillations. This paper presents a novel CV-based high-resolution limiter to effectively suppress oscillations and maintain CV resolution. Drawing inspiration from the SWENO method [43], we utilize a nonlinear weighting approach to reconstruct a novel high-order polynomial for the target control volume by combining the high-order reconstructed polynomial and linear polynomials which are reconstructed by the cell average of the target CV and its neighboring CVs. The new high-order polynomial breaks the continuity in the SV, thus the utilization of numerical flux at the boundaries of troubled CVs and the SV boundaries. However, at other boundaries of CVs where physical quantities remain continuous, direct calculation of flux based on the values of physical variables is feasible. The limiter is simple, as it only requires several linear polynomials in the limitation process. Moreover, it still maintains the compactness of the SV method and preserves the resolution of CV. Numerical results for one- and two-dimensional scalar and system of conservation laws verified that the high-order property of the CV-SWENO limiter is effective in solving both smooth and strongly discontinuity problems.

Figures (15)

• Figure 4.1: Example \ref{['chap03:sod1d']}: One-dimensional Sod shock tube problem: When $t=2$ and $M=0.01$,the density numerical solution is obtained using 100 spectral volumes in SV-cvMSWNO3-SV-cvMSWENO5 scheme. The solid line and symbol "$\square$" represent the exact solution and SV-cvMSWENO results, respectively.
• Figure 4.2: Example \ref{['chap03:sod1d']}: One-dimensional Sod shock tube problem: The density numerical solution obtained using 100 spectral volume elements in SV-cvMSWNO3-SV-cvMSWENO5 scheme with different $M$ values at $t=2$. The solid line and symbol "$\square$" represent the exact solution and SV-cvMSWENO results, respectively.
• Figure 4.3: Example \ref{['chap03:lax1d']}: One-dimensional Lax shock tube problem: The density numerical solution obtained using 100 spectral volume elements in SV-cvMSWNO3-SV-cvMSWENO5 scheme when $M=0.01$ at $t=1.3$. The solid line and symbol "$\square$" represent the exact solution and SV-cvMSWENO results, respectively.
• Figure 4.4: Example \ref{['chap03:lax1d']}: One-dimensional Lax shock tube problem: The density numerical solution obtained using 100 spectral volumes when fully restricted by SV-cvMSWNO3-SV-cvMSWENO5 scheme at $t=1.3$. The solid line and symbol "$\square$" represent the exact solution and SV-cvMSWENO results, respectively.
• Figure 4.5: Example \ref{['chap03:shuosher']}: Shock wave and sine wave interaction problem: The density numerical solution obtained using 180 spectral volumes in SV-cvMSWNO3-SV-cvMSWENO5 scheme when $M=0.01$ at $t=1.8$. The solid line and symbol "$\square$" represent the exact solution and SV-cvMSWENO results, respectively.

Theorems & Definitions (11)

• Remark 2.1
• Remark 3.1
• Example 4.1: Accuracy test of the one-dimensional linear convection equation
• Example 4.2: Precision test of the one-dimensional Euler equations
• Example 4.3: One-dimensional Sod shock tube problem
• Example 4.4: One-dimensional Lax shock tube problem
• Example 4.5: Shock and sine wave interaction problem
• Example 4.6: Explosive wave problem
• Example 4.7: 2D Riemannproblem I
• Example 4.8: 2D Riemannproblem II