Efficient solution of non-Markovian covariance evolution equations in fluid turbulence

Abstract

A method is described for efficient treatment of the time-history effects that arise in closure models constructed by truncation of renormalized perturbation expansions. The scheme is based on rational polynomial (Pade) approximations for the time-lagged functions, combined with Gaussian quadrature applied to a coarse-time evolution equation for equal-time covariances. Power series coefficients required for the Pade approximants are generated by recursion relations resulting from successive differentiation of the slow-varying form of the dynamic equations for time-lagged functions. This strategy is illustrated by applications to equations of the direct interaction approximation for fluid turbulence. Several benefits accrue from such an approach. Computational efficiency comparable to that of Markovian closures is obtained without introduction of arbitrary constants orad hoc Markovianization assumptions: (1) Scaling of the leading operations count is reduced fromO(n T 3 ) toO(n g n t ), wheren T is the number of steps in coarse time, and ng≈ 4–8. For typical problems in fluid turbulence, this results in savings of afactor∼40 in computing time. (2) Scaling of working array size is reduced fromO(n T 2 ) toO(p), wherep is the order of Pade approximant; this reduces in-core storage requirements for dynamical quantities by afactor ∼20. These and other advantages depend on the order of Pade approximant required to give acceptable accuracy. Test cases from several problems in fluid turbulence indicate very substantial savings, such that certain CPU-intensive problems with spatial inhomogeneities become computationally feasible.