Comparison of highly accurate interpolation methods

Abstract

The accuracy of two general interpolation methods was investigated within the context of developing a high-order overset grid flow solver. The two methods, a generalized Lagrangian method and a basisspline method, were examined using a one-dimensional Fourier error analysis. The local and integrated error was reduced for both implicit and explicit Lagrangian methods by optimizing the coefficients at the expense of reducing formal order of accuracy. For the stencil sizes investigated here, the optimized implicit interpolation methods were shown to possess integrated errors many orders of magnitude lower than the basis-spline, non-optimized or optimized explicit methods. The generalized Lagrangian interpolation method was chosen for implementation with an established high-order flow solver to produce a high-order overset grid capability. A two-dimensional benchmark problem of an inviscid, connecting vortex was solved using this capability on an overset grid system consisting of both Cartesian and curvilinear grids. Initial results showed substantial improvement in the computed solution for the optimized implicit interpolation method when compared to the bilinear interpolation method. The optimized implicit method reduced the maximum difference between the computed and exact solutions by a factor of 9.4 when compared to the bilinear method. INTRODUCTION High-order compact finite-difference schemes continue to mature as a means of simulating various wavepropagation phenomena. Recent works by Rizzetta et * Research Aerospace Engineer. Member AIAA '''Associate Professor. Associate Fellow AIAA This paper is a -work of the U.S. Government and is not subject to copyright protection in the United States. al.' in the field of turbulent flow simulation, Visbal and Gaitonde in aeroacoustics, and Gaitonde et al.' and Shang in electromagnetics have demonstrated the ability to solve increasingly complex multidisciplinary problems using high-order methods on generalized curvilinear grids. These methods are based on Lele's Pade-type, implicit tridiagonal, fourthand sixth-order compact finite differences. Because these schemes are centered, they are non-dissipative hi nature, and thus high-order filters are employed to remove spurious frequencies from the solution. These filters include an adjustable parameter that controls spectral response by shifting the frequency cut-off to higher or lower wavenumbers as well as modifying its sharpness.' Thus, the characteristics of the filter may be easily modified depending upon the quality and resolution of the computational grid as well as the inherent nonlinearities or other destabilizing features of the particular problem being examined. This technology was further developed by Gaitonde and Visbal'' by introducing high-order, one-sided filters to maintain accuracy and stability near boundaries. This enhancement allowed the extension of the approach to multi-domain problems with coincident, overlapping grid points at the domain interfaces. Overset grid methods also continue to develop as an effective means of simulating physical phenomenon for geometrically complex problems or problems with multiple bodies in relative motion. Blake and Shang used an overset grid approach to simulate electromagnetic scattering from a full B-1B aircraft, while Chan and Gomez have developed an automated overset grid generation procedure and examined surface pressures on an X-38 Crew Return Vehicle. Meakin, Meakin and Wissink, and Wang et al. have investigated issues and solved complex flow problems