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

TL;DR

This paper introduces a simple, high-order, compact WENO limiter based on control volume principles for spectral volume methods, effectively reducing oscillations near discontinuities while maintaining high resolution.

Contribution

It proposes a novel CV-based high-resolution limiter inspired by SWENO, combining high-order and linear polynomials for improved oscillation suppression in spectral volume methods.

Findings

Effective suppression of oscillations in discontinuous problems

Maintains high-order accuracy and resolution

Works well for scalar and system conservation laws

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.