Jae-Hun Jung

Jingyang Guo, Jae-Hun Jung

The essentially non-oscillatory (ENO) method is an efficient high order numerical method for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations, if existent, by adaptively choosing the local stencil for the interpolation. The original ENO method is constructed based on the polynomial interpolation and the overall rate of convergence provided by the method is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose simple non-polynomial ENO and weighted ENO (WENO) finite volume methods in order to enhance the local accuracy and convergence. We first adopt the infinitely smooth radial basis functions (RBFs) for a non-polynomial interpolation. Particularly we use the multi-quadric and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. That is, it is not necessary to know the exact form of the non-polynomial basis for the interpolation. In this paper, we formulate the ENO and WENO methods based on the non-polynomial interpolation and derive the optimization condition of the perturbation. To guarantee the essentially non-oscillatory property, we switch the non-polynomial reconstruction to the polynomial reconstruction adaptively near the non-smooth area by using the monotone polynomial interpolation method. The numerical results show that the developed non-polynomial ENO and WENO methods enhance the local accuracy.

Kwanghyuk Park, Jiaxi Gu, Jae-Hun Jung

In this work, we present the feedforward neural network based on the conservative approximation to the derivative from point values, for the weighted essentially non-oscillatory (WENO) schemes in solving hyperbolic conservation laws. The feedforward neural network, whose inputs are point values from the three-point stencil and outputs are two nonlinear weights, takes the place of the classical WENO weighting procedure. For the training phase, we employ the supervised learning and create a new labeled dataset for one-dimensional conservative approximation, where we construct a numerical flux function from the given point values such that the flux difference approximates the derivative to high-order accuracy. The symmetric-balancing term is introduced for the loss function so that it propels the neural network to match the conservative approximation to the derivative and satisfy the symmetric property that WENO3-JS and WENO3-Z have in common. The consequent WENO schemes, WENO3-CADNNs, demonstrate robust generalization across various benchmark scenarios and resolutions, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.