On the efficiency and accuracy of interpolation methods for spectral codes

In this paper a general theory for interpolation methods on a rectangular grid is introduced. By the use of this theory an efficient B-spline based interpolation method for spectral codes is presented. The theory links the order of the interpolation method with its spectral properties. In this way many properties like order of continuity, order of convergence and magnitude of errors can be explained. Furthermore, a fast implementation of the interpolation methods is given. We show that the B-spline based interpolation method has several advantages compared to other methods. First, the order of continuity of the interpolated field is higher than for other methods. Second, only one FFT is needed whereas e.g. Hermite interpolation needs multiple FFTs for computing the derivatives. Third, the interpolation error almost matches the one of Hermite interpolation, a property not reached by other methods investigated.Comment: 19 pages, 5 figure

On optimal interpolation schemes for particle tracking in turbulence

An important aspect in numerical simulations of particle-laden turbulent flows is the interpolation of the flow fleld needed for the computation of the Lagrangian trajectories. The accuracy of the interpolation method has direct consequences for the acceleration spectrum of the fluid particles and is therefore also important for the correct evaluation of the hydrodynamic forces for almost neutrally buoyant particles, common in many environmental applications. In order to systematically choose the optimal tradeoff between interpolation accuracy and computational cost we focuss on comparing errors: the interpolation error is compared with the discretisation error of the flow field. In this way one can prevent unnecessary computations and still retain the accuracy of the turbulent flow simulation. From the analysis a practical method is proposed that enables direct estimation of the interpolation and discretization error from the energy spectrum. The theory is validated by means of Direct Numerical Simulations (DNS) of homogeneous, isotropic turbulence using a spectral code, where the trajectories of fluid tracers are computed using several interpolation methods. We show that B-spline interpolation has the best accuracy given the computational cost. Finally, the optimal interpolation order for the different methods is shown as a function of the resolution of the DNS simulation

Two hierarchies of spline interpolations. Practical algorithms for multivariate higher order splines

A systematic construction of higher order splines using two hierarchies of polynomials is presented. Explicit instructions on how to implement one of these hierarchies are given. The results are limited to interpolations on regular, rectangular grids, but an approach to other types of grids is also discussed

Application of the nonuniform fast Fourier transform to the direct numerical simulation of two-way coupled particle laden flows

We present the application of the Nonuniform Fast Fourier Transform (NUFFT) to the pseudo-spectral Eulerian–Lagrangian direct numerical simulation of particle-laden flows. In the two-way coupling regime, when the particle feedback on the flow is taken into account, a spectral method requires not only the interpolation of the flow fields at particle positions, but also the Fourier representation of the particle back-reaction on the flow fields on a regular grid. Even though the direct B-spline interpolation is a well-established tool, to the best of our knowledge the reverse projection scheme has never been used, replaced by less accurate linear reverse interpolation or Gaussian regularization. We propose to compute the particle momentum and temperature feedback on the flow by means of the forward NUFFT, while the backward NUFFT is used to perform the B-spline interpolation. Since the backward and forward transformations are symmetric and the (non local) convolution computed in physical space is removed in Fourier space, this procedure satisfies all constraints for a consistent interpolation scheme, and allows an efficient implementation of high-order interpolations. The resulting method is applied to the direct numerical simulation of a forced and isotropic turbulent flow with different particle Stokes numbers in the two-way coupling regime. A marked multifractal scaling is observed in the particle statistics, which implies that the feedback from the particles on the fields is far from being analytic and therefore only high-order methods, like the one here proposed, can provide an accurate representation

Persistent accelerations disentangle Lagrangian turbulence

Particles in turbulence frequently encounter extreme accelerations between extended periods of quiescence. The occurrence of extreme events is closely related to the intermittent spatial distribution of intense flow structures such as vorticity filaments. This mixed history of flow conditions leads to very complex particle statistics with a pronounced scale dependence, which presents one of the major challenges on the way to a non-equilibrium statistical mechanics of turbulence. Here, we introduce the notion of persistent Lagrangian acceleration, quantified by the squared particle acceleration coarse-grained over a viscous time scale. Conditioning Lagrangian particle data from simulations on this coarse-grained acceleration, we find remarkably simple, close-to-Gaussian statistics for a range of Reynolds numbers. This opens the possibility to decompose the complex particle statistics into much simpler sub-ensembles. Based on this observation, we develop a comprehensive theoretical framework for Lagrangian single-particle statistics that captures the acceleration, velocity increments as well as single-particle dispersion

A divergence-free constrained magnetic field interpolation method for scattered data

An interpolation method to evaluate magnetic fields given unstructured, scattered magnetic data is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L^2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constrain in the cost function allowing the method to achieve arbitrarily small errors in the magnetic field divergence. An exponential decay of the approximation error is observed and compared with the less favorable algebraic decay of local splines. Compared to local methods involving computationally expensive search algorithms, the proposed method exhibits a significant reduction of the computational complexity of the field evaluation, while maintaining a small error in the divergence even in the presence of magnetic islands and stochasticity. Applications to the computation of Poincar\'e sections using data obtained from numerical solutions of the magnetohydrodynamic equations in toroidal geometry are presented and compared with local methods currently in use