Intelligent denoising methods for Gibbs phenomenon of high-order discontinuous Galerkin numerical scheme

The high-order discontinuous Galerkin (DG) method for the high-speed compressible flow field calculations will cause non-physical numerical oscillation near the shock wave, affecting numerical accuracy and even leading to calculation failure, similar to the accumulation of Gibbs noise in the image processing field. In the high-order DG methods, restraining shock oscillation or eliminating the Gibbs phenomenon to ensure the stability of the calculation process has become a challenge. A Gibbs phenomenon intelligent denoising model composed of graph attention mechanism and graph convolutional network is proposed by using machine learning technology. This model can restrain the oscillation near the shock wave in DG calculation, while ensuring the convergence of DG calculation and improving the effectiveness of shock wave capturing. After constructing a training dataset from Gibbs noise data generated by DG calculation, this model trains the graph neural network under the guidance of graph convolutional filters. In the numerical simulation experiment of NACA0012 airfoil under transonic and supersonic inflow conditions, the Gibbs phenomenon intelligent denoising model is embedded in the DG calculation. The experimental results show that the Gibbs phenomenon has been eliminated and shock oscillation has been effectively restrained .