Qing-Dong Cai

Ling Wen, Yan-Tao Yang, Qing-Dong Cai

This paper presents a novel and straightforward compact reconstruction procedure for the high-order finite volume method on unstructured grids. In this procedure, we constructed a linear approximation relationship between the mean values and the function values, as well as the derivative values. Compared with the classical compact schemes, which employ a Taylor expansion method to determine the coefficients, our approach adopts an equivalent and more generalized method to achieve this goal. Via this method, the problem of constructing a high-order compact scheme is transformed into solving the null space of undetermined homogeneous linear systems. This null space constitutes the complete set of schemes that meet the specified accuracy under a given stencil, and is termed the 'scheme space'. Schemes within the scheme space possess the same accuracy level yet exhibit distinct dispersion and dissipation characteristics. Through Fourier analysis, we can get the dissipation and dispersion properties of all schemes in the scheme space. This facilitates the control of scheme dispersion and dissipation without altering the stencil compactness. Combined with the WENO (Weighted Essentially Non-Oscillatory) concept, multi-stencil schemes are employed to construct the nonlinear weighted compact finite volume scheme (WCFV). The WCFV is capable of eliminating unphysical oscillations at discontinuities, thereby enabling the capture of strong discontinuities. One-dimensional schemes are discussed in detail, and numerical results demonstrate that the proposed method exhibits high-order accuracy, robustness, and shock-capturing capability.

Zeyu Liu, Yantao Yang, Qing-Dong Cai

In this paper, we present a novel framework to solve differential equations based on multilayer feedforward network. Previous works indicate that solvers based on neural network have low accuracy due to that the boundary conditions are not satisfied accurately. The boundary condition is now inserted directly into the model as boundary term, and the model is a combination of a boundary term and a multilayer feedforward network with its weight function. As the boundary condition becomes predefined constraintion in the model itself, the neural network is trained as an unconstrained optimization problem. This leads to both ease of training and high accuracy. Due to universal convergency of multilayer feedforward networks, the new method is a general approach in solving different types of differential equations. Numerical examples solving ODEs and PDEs with Dirichlet boundary condition are presented and discussed.