In this work, we develop a novel energy-stable linear approach, which we name as auxiliary variable viscosity splitting (AVVS) method, to efficiently solve the incompressible fluid flows. Different from the projection-type methods with pressure correction, the AVVS method adopts the viscosity splitting strategy to split the original momentum equation into an intermediate momentum equation without divergence-free constraint and an advection-free momentum equation. A time-dependent auxiliary variable which has exact value 1 is introduced to construct a supplementary equation. The new model not only inherits the same dynamics of original incompressible Navier–Stokes equations, but also facilitates us to design linearly decoupled and energy-stable time-marching scheme. Comparing with the conventional projection-type schemes, the present method leads to an energy dissipation law with respect to kinetic energy instead of a modified energy including velocity and pressure gradient. In each time step, only two parabolic equations with constant coefficients and one Poisson equation need to be solved. Therefore, the numerical implementation is highly efficient. Moreover, the proposed AVVS method can be directly extended to construct linear, decoupled, and energy-stable scheme for the turbidity current system with slight modifications on the right-hand side of supplementary equation. Extensive numerical experiments are implemented to validate the accuracy, energy stability, and capability in complex fluid simulations.

中文翻译：

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不可压缩纳维-斯托克斯方程和浊流系统的能量稳定辅助变粘度分裂（AVVS）方法


在这项工作中，我们开发了一种新颖的能量稳定线性方法，我们将其命名为辅助变粘度分裂（AVVS）方法，以有效地解决不可压缩流体流动。与带有压力修正的投影型方法不同，AVVS方法采用粘性分裂策略，将原始动量方程分裂为无散度约束的中间动量方程和无平流动量方程。引入精确值为 1 的与时间相关的辅助变量来构造补充方程。新模型不仅继承了原始不可压缩纳维-斯托克斯方程的相同动力学特性，而且有助于我们设计线性解耦和能量稳定的时间推进方案。与传统的投影型方案相比，本方法得出关于动能的能量耗散定律，而不是包括速度和压力梯度的修正能量。在每个时间步中，只需要求解两个常系数抛物线方程和一个泊松方程。因此，数值实现的效率很高。此外，所提出的AVVS方法可以直接扩展为浊流系统构造线性、解耦和能量稳定的方案，只需对补充方程右侧稍加修改即可。进行了大量的数值实验来验证复杂流体模拟的准确性、能量稳定性和能力。