Abstract
A method is proposed for the derivation of new classes of staggered compact derivative and interpolation operators. The algorithm has its roots in an implicit interpolation theory consistent with compact schemes and reduces to the computation of the required staggered derivatives and interpolated quantities as a combination of nodal values and collocated compact derivatives. The new approach is cost-effective, simplifies the imposition of boundary conditions, and has improved spectral resolution properties, on equal order of accuracy, with respect to classical schemes. The method is applied to incompressible Navier-Stokes equations through the implementation into a staggered flow solver with a fractional step procedure, and tested on classical benchmarks.
Highlights
Direct numerical simulation of many complex multi-scale problems, such as turbulence in fluids, requires the use of adequately refined grids to resolve the structure and the dynamics of the smallest scales
They showed, for example, that the classical Padé fourth-order formula for first derivative can be obtained by analytically differentiating the cubic spline interpolation, recognizing that compact schemes could be derived through a theory of polynomial interpolation in physical space, which has to be necessarily implicit
We present a strategy for the generation of several compact derivatives starting from the computation of a single set of compact schemes
Summary
Direct numerical simulation of many complex multi-scale problems, such as turbulence in fluids, requires the use of adequately refined grids to resolve the structure and the dynamics of the smallest scales. In a series of papers, Rubin and co-workers [8,9,10] firstly estabilished that many compact schemes can be derived by employing the theory of spline interpolation They showed, for example, that the classical Padé fourth-order formula for first derivative can be obtained by analytically differentiating the cubic spline interpolation, recognizing that compact schemes could be derived through a theory of polynomial interpolation in physical space, which has to be necessarily implicit. They obtained formulæ for second derivatives and for the case of nonuniform mesh, and applied their schemes to a variety of fluid flow problems.
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