A quadratic conservation algorithm for non-hydrostatic models

We propose a quadratic conservation algorithm for non-hydrostatic models, focusing on energy conservation and consistency. To ensure that the discrete potential and continuity equations are equivalent and to achieve conservation of mass and momentum transport, we introduce a potential reference height into the discrete potential equation. The time integration scheme is based on an explicit fourth-order Runge-Kutta method with varying time steps, which is specially designed to maintain the quadratic conservation property. To suppress numerical noise, we suggest an adaptive diffusion method that does not lose energy. Numerical noise is estimated by higher order terms of polynomial equations, and diffusion coefficients are determined using a least-squares method. Numerical tests demonstrate that the quadratic conservation algorithm accurately maintains total energy conservation and produces solutions of comparable quality to those reported in existing literature. Furthermore, it can resolve problems with small-scale features. The diagram illustrates the visual representation of the grid obtained by cutting along the line x=i Delta x$$ x=i\Delta x $$, as defined by Equation (14). The bottom and top boundary conditions are defined in the regions embedded with equations, which ensure that the flux and diffusion from the regions are always zero. Therefore, the regions are not necessary for actual computations. The formulas on the right side of the grid constitute the stencil of Equation (16), and the solid dots indicate their spatial positions. image