We present a filter stabilization technique for the mildly compressible Euler equations that relies on a linear or nonlinear indicator function to identify the regions of the domain where artificial viscosity is needed and determine its amount. For the realization of this technique, we adopt a three step algorithm called Evolve-Filter-Relax (EFR), which at every time step evolves the solution (i.e., solves the Euler equations on a coarse mesh), then filters the computed solution, and finally performs a relaxation step to combine the filtered and non-filtered solutions. We show that the EFR algorithm is equivalent to an eddy-viscosity model in Large Eddy Simulation. Three indicator functions are considered: a constant function (leading to a linear filter), a function proportional to the norm of the velocity gradient (recovering a Smagorinsky-like model), and a function based on approximate deconvolution operators. Through well-known benchmarks for atmospheric flow, we show that the deconvolution-based filter yields stable solutions that are much less dissipative than the linear filter and the Samgorinsky-like model and we highlight the efficiency of the EFR algorithm.

翻译：我们针对适度可压缩欧拉方程提出一种滤波稳定化技术，该技术依赖线性或非线性指示函数来识别需要引入人工黏性的区域并确定其强度。为实现该技术，我们采用名为"演化-滤波-松弛"（EFR）的三步算法：每个时间步先对解进行演化（即在粗网格上求解欧拉方程），接着对计算所得解进行滤波，最后通过松弛步骤将滤波解与未滤波解进行组合。我们证明EFR算法等价于大涡模拟中的涡黏模型。研究考虑三种指示函数：常数函数（生成线性滤波器）、正比于速度梯度范数的函数（恢复Smagorinsky类模型），以及基于近似反卷积算子的函数。通过大气流动的经典基准算例，我们证明基于反卷积的滤波器能产生稳定解，且其耗散性远低于线性滤波器和Smagorinsky类模型，同时凸显了EFR算法的高效性。