LAUSR

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.