An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations

Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much more computational cost and thus is not suite for large scale simulation such as direct numeric simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO scheme, i.e. the AdaWENO scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. The new scheme performs characteristic-wise reconstruction near discontinuities while switching to component-wise reconstruction for smooth regions. Meanwhile, a new calculation strategy for the WENO smoothness indicators is implemented to reduce over-all computational cost. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO scheme. Numerical results show that AdaWENO maintains essentially non-oscillatory flow field near discontinuities as the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction, AdaWENO is about 40\% faster which indicates its excellent efficiency

Enhancement of shock-capturing methods via machine learning

In recent years, machine learning has been used to create data-driven solutions to problems for which an algorithmic solution is intractable, as well as fine-tuning existing algorithms. This research applies machine learning to the development of an improved finite-volume method for simulating PDEs with discontinuous solutions. Shock-capturing methods make use of nonlinear switching functions that are not guaranteed to be optimal. Because data can be used to learn nonlinear relationships, we train a neural network to improve the results of a fifth-order WENO method. We post-process the outputs of the neural network to guarantee that the method is consistent. The training data consist of the exact mapping between cell averages and interpolated values for a set of integrable functions that represent waveforms we would expect to see while simulating a PDE. We demonstrate our method on linear advection of a discontinuous function, the inviscid Burgers’ equation, and the 1-D Euler equations. For the latter, we examine the Shu–Osher model problem for turbulence–shock wave interactions. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused due to numerical viscosity