A Fourth-order Conservative Adaptive Multiresolution Wavelet Upwind Scheme for Compressible Flows

Abstract: A fourth-order conservative adaptive multiresolution average-interpolating wavelet upwind scheme is proposed for compressible flows governed by hyperbolic conservation laws. A family of asymmetric average-interpolating wavelets with upwind properties is constructed for conservative finite volume discretization, while symmetric average-interpolating wavelets are employed for multiresolution decomposition and reconstruction of physical variables in the adaptive procedure. Since both the conservative discretization and the adaptive multiresolution representation are constructed from cell-average quantities, the proposed scheme preserves strict conservation during both numerical evolution and adaptive cell redistribution. Unlike hybrid adaptive wavelet methods that use wavelets mainly for data compression and mesh adaptation, the present adaptive wavelet upwind scheme utilizes average-interpolating wavelet multiresolution approximation to reconstruct the interface values directly for numerical flux evaluation, thereby avoiding additional ghost-cell marking and reconstruction near coarse--fine mesh interfaces. The boundary variation diminishing reconstruction is incorporated at the finest resolution level to achieve non-oscillatory shock-capturing capability. Numerical tests demonstrate that the proposed scheme achieves the expected fourth-order accuracy, maintains conservation errors close to machine precision, and controls numerical errors around the prescribed threshold. The proposed method also sharply captures shock waves and contact discontinuities without spurious oscillations and resolves multiscale smooth structures through a sparse adaptive representation. These results indicate that the proposed scheme provides an efficient, conservative, and reliable approach for high-resolution simulations of compressible flows.