Last modified: 2024-05-12
Abstract
High-speed flows governed by hyperbolic conservation laws often involve steep gradients or even discontinuities, such as shock waves and contact discontinuities, irrespective of the smoothness of the initial and boundary conditions. Meanwhile, smooth structures, e.g., multiscale vortices and acoustic waves, are also included in the solutions. Hence, it is still a challenging task to design a satisfactory numerical method to resolve the hyperbolic conservation laws efficiently.
Although traditional mesh-based high-order schemes can capture the discontinuities without spurious oscillations and distinguish different scale smooth structures successfully, they are not cost-effective since fine meshes are necessary to resolve these localized structures. Multiresolution analysis and time-frequency localization characteristics of wavelet theory provide a potential tool to devise adaptive multiresolution wavelet methods with high-order accuracy for efficiently addressing the hyperbolic conservation laws. Inspired by the classical upwind schemes, we introduce a novel system of adaptive multiresolution wavelet collocation upwind schemes based on second-generation wavelets, leveraging the properties of different wavelets. In the proposed schemes, weak asymmetrical interpolating wavelets ensuring the upwind property are used to discretize the hyperbolic conservation laws spatially. Symmetrical second-generation wavelets are utilized to decompose and reconstruct the solutions in the adaptive process, preserving the shape of the solution and achieving better efficiency. In addition, our schemes conduct the direct discretization of relevant spatial derivatives in the physical space without local characteristics decomposition, which further improves their efficiency.
Several numerical tests are conducted to verify the enhanced performance of these schemes. The proposed schemes can achieve similar solutions as the interpolating wavelet upwind schemes and the classical WENO5 scheme by employing much fewer nodes and saving large computational costs. All numerical tests demonstrate that the newly proposed schemes can efficiently resolve the hyperbolic conservation laws with a higher data compression rate.