A quadratic conservation algorithm for non‐hydrostatic models

Abstract

AbstractWe 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.