The shallow water equations with source term have steady-state solution. The numerical scheme for solving this kind of equations must have well-balanced property, otherwise it will cause oscillation. For the bottom-non-flat shallow water with source term, this paper studies the well-balanced property of the high order compact central weighted essentially non-oscillatory (CCWENO) entropy stable scheme, and proves its well-balanced property. The theory is verified by one- and two-dimensional numerical examples. The numerical results show that the high order CCWENO scheme has the well-balanced property and can accurately capture the small perturbation of the solution even based on the coarse grid.

Because of the shortcomings of classical PINN (Physical-informed Neural Networks) for discontinuous problems of shallow water equation, a regularized PINN algorithm based on viscous dissipative mechanism was proposed. In the network framework, the viscous regularized shallow water equation is used as the physical constraint and the penalty term in the loss function. Training network makes the smooth solution of the regularized equation approximate the discontinuous solution of the original equation. Finally, for one-dimensional and two-dimensional shallow water problems with different initial conditions, the numerical results show that the new algorithm has strong generalization ability, can predict the solution at any time, and has high resolution, without the phenomenon of spurious oscillation.

In this paper we give a sufficient condition for sign preservation in Weighted Essentially Non-Oscillatory with Adaptive Order (WENO-AO) reconstruction, which means to keep same sign for jumping at interfaces. WENO-AO reconstruction realizes self-adaptive accuracy through an nonlinear combination of high-order polynomial and low-order reconstruction. In solving solution near discontinuous points, WENO-AO reconstruction is more accurate than the classical WENO reconstruction, and it keeps good stability. The scheme is obtained by combining a central upwind numerical flux with a third-order strongly stable Runge-Kutta method. Numerical results show that the scheme has fifth order accuracy. It has characteristics of high resolution and strong robustness. It captures accurately discontinuous positions and avoids effectively the generation of spurious oscillation.

A high resolution entropy consistent scheme for ideal MHD equations is presented. A slope limiter based on MUSCL-type data reconstruction method is constructed with analysis on conditions required for non-oscillation. The scheme has high accuracy in the smooth region of the solution. In the discontinuous region the numerical dissipation can be controlled reasonably and the non-physical phenomena can be avoided effectively. The entropy stable scheme, the entropy consistent scheme and the new high-resolution entropy consistent scheme are used to simulate one-dimensional and two-dimensional ideal MHD equations. It shows that the new scheme captures structure of the solution accurately, and has the characteristics of non-oscillation, high-resolution and robustness.