Abstract The simulation of multiscale problems remains a challenge in many problems due to the complex interaction between the resolved and unresolved scales. This work develops a coarse-grained modeling approach for Galerkin discretizations by combining the Variational Multiscale decomposition and the Mori–Zwanzig (M–Z) formalism. An appeal of the M–Z formalism is that – akin to Green’s functions for linear problems – the impact of unresolved dynamics on resolved scales can be formally represented as a convolution (or memory) integral in a non-linear setting. To ensure tractable and efficient models, Markovian closures are developed for the M–Z memory integral. The resulting sub-scale model has some similarities to adjoint stabilization and orthogonal sub-scale models. The model is made parameter-free by adaptively determining the memory length during the simulation. To illustrate the generalizability of this model, the performance is assessed in detail in coarse-grained simulations of a range of problems from the one-dimensional Burgers equation and to incompressible turbulence.

中文翻译：

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使用 Mori-Zwanzig 方法的有限元离散化的变分多尺度闭包

摘要 由于已解决和未解决的尺度之间复杂的相互作用，多尺度问题的模拟在许多问题中仍然是一个挑战。这项工作通过结合变分多尺度分解和 Mori-Zwanzig (M-Z) 形式主义，为 Galerkin 离散化开发了一种粗粒度建模方法。M-Z 形式主义的一个吸引力在于——类似于用于线性问题的格林函数——未解析动态对已解析尺度的影响可以正式表示为非线性设置中的卷积（或记忆）积分。为确保模型易于处理和高效，为 M-Z 记忆积分开发了马尔可夫闭包。由此产生的子尺度模型与伴随稳定和正交子尺度模型有一些相似之处。通过在仿真过程中自适应地确定内存长度，使模型无参数。为了说明该模型的通用性，在从一维 Burgers 方程到不可压缩湍流的一系列问题的粗粒度模拟中详细评估了性能。