Yohsuke Imai

Accuracy study of the IDO scheme by Fourier analysis

The numerical accuracy of the Interpolated Differential Operator (IDO) scheme is studied with Fourier analysis for the solutions of Partial Differential Equations (PDEs): advection, diffusion, and Poisson equations. The IDO scheme solves governing equations not only for physical variable but also for first-order spatial derivative. Spatial discretizations are based on Hermite interpolation functions with both of them. In the Fourier analysis for the IDO scheme, the Fourier coefficients of the physical variable and the first-order derivative are coupled by the equations derived from the governing equations. The analysis shows the IDO scheme resolves all the wavenumbers with higher accuracy than the fourth-order Finite Difference (FD) and Compact Difference (CD) schemes for advection equation. In particular, for high wavenumbers, the accuracy is superior to that of the sixth-order Combined Compact Difference (CCD) scheme. The diffusion and Poisson equations are also more accurately solved in comparison with the FD and CD schemes. These results show that the IDO scheme guarantees highly resolved solutions for all the terms of fluid flow equations.

Stable coupling between vector and scalar variables for the IDO scheme on collocated grids

The Interpolated Differential Operator (IDO) scheme on collocated grids provides fourth-order discretizations for all the terms of the fluid flow equations. However, computations of fluid flows on collocated grids are not guaranteed to produce accurate solutions because of the poor coupling between velocity vector and scalar variables. A stable coupling method for the IDO scheme on collocated grids is proposed, where a new representation of first-order derivatives is adopted. It is important in deriving the representation to refer to the variables at neighboring grid points, keeping fourth-order truncation error. It is clear that accuracy and stability are drastically improved for shallow water equations in comparison with the conventional IDO scheme. The effects of the stable coupling are confirmed in incompressible flow calculations for DNS of turbulence and a driven cavity problem. The introduction of a rational function into the proposed method makes it possible to calculate shock waves with the initial conditions of extreme density and pressure jumps.