A conservative wavelet upwind scheme for compressible flows

In this paper, we develop a fourth-order conservative wavelet-based shock-capturing scheme. The scheme is constructed by combining a wavelet collocation upwind method with the monotonic tangent of hyperbola for interface capturing (THINC) technique. We employ boundary variation diminishing (BVD) reconstruction to enhance the scheme’s effectiveness in handling shocks. First, we prove that wavelet collocation upwind schemes based on interpolating wavelets can be reformulated into a conservative form within the framework of wavelet theory, forming the foundation of the proposed scheme. The new fourth-order accurate scheme possesses significantly better spectral resolution than the fifth- and even seventh-order WENO-Z (weighted essentially non-oscillatory) schemes over the entire wave-number range. Moreover, the inherent low-pass filtering property of the wavelet bases allows them to filter high-frequency numerical oscillations, endowing the wavelet upwind scheme with robustness and accuracy in solving problems under extreme conditions. Notably, due to the wavelet multi-resolution approximation, the proposed scheme possesses a distinctive shape-preserving property absent in the WENO-Z schemes and the fifth-order schemes with BVD reconstruction based on polynomials. Furthermore, compared to the fifth-order scheme with BVD reconstruction based on polynomials—which is significantly superior to the WENO schemes—the proposed scheme further enhances the ability to capture discontinuities.