Efficient nonlinear filter stabilization of the Leray-α model

Abstract

In this work we study a novel adaptive nonlinear filtering applied to the Leray-α model. Unlike its classical counterpart, the new filtering requires the solution of a linear elliptic problem, with constant coefficients, at each time step. The action of the adaptive nonlinear filter throughout the integration time is refactorized as the solution of a linear system, with the same matrix and multiple right hand-sides. We discuss the theoretical properties of the new filtering approach applied to the BDF2 approximation of the Leray-α model. Theoretical results and numerical tests demonstrate that the linear system arising from spatial discretization is well-conditioned and the accuracy of the filter is comparable with the classical one. Some benchmark results are also presented.