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Accuracy study of the IDO scheme by Fourier analysis. (English) Zbl 1102.65094
Summary: The numerical accuracy of the Interpolated Differential Operator (IDO) scheme is studied with Fourier analysis for the solutions of partial differential equations: 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 wave-numbers with higher accuracy than the fourth-order finite difference (FD) and compact difference (CD) schemes for the advection equation. In particular, for high wave-numbers, the accuracy is superior to that of the sixth-order combined compact difference 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.

##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35K05 Heat equation 35L45 Initial value problems for first-order hyperbolic systems
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##### References:
 [1] Gottlieb, D.; Orszag, A., Numerical analysis of spectral methods, (1977), SIAM Philadelphia, PA · Zbl 0412.65058 [2] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed turbulent channel flow at low Reynolds number, J. fluid mech., 177, 133, (1987) · Zbl 0616.76071 [3] Spalart, P., Direct numerical simulation of a turbulent boundary layer up to reθ=1410, J. fluid mech., 187, 61, (1988) · Zbl 0641.76050 [4] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006 [5] Yabe, T.; Aoki, T., A universal solver for hyperbolic equations by cubic-polynomial interpolation I. one-dimensional solver, Comput. phys. commun., 66, 219, (1991) · Zbl 0991.65521 [6] Yabe, T.; Xiao, F.; Utsumi, T., The constrained interpolation profile method for multiphase analysis, J. comput. phys., 169, 556, (2001) · Zbl 1047.76104 [7] Utsumi, T.; Aoki, T.; Kunugi, T., Stability and accuracy of the cubic interpolated propagation scheme, Comput. phys. commun., 101, 9, (1997) [8] Tanaka, R.; Nakamura, T.; Yabe, T., Constructing exactly conservative scheme in a non-conservative form, Comput. phys. commun., 126, 232, (2000) · Zbl 0959.65097 [9] Takizawa, K.; Yabe, T.; Nakamura, T., Multi-dimensional semi-Lagrangian scheme that guarantees exact conservation, Comput. phys. commun., 148, 137, (2002) · Zbl 1196.65147 [10] Xiao, F., A simple CIP finite volume method for incompressible flows, JSME int. J. B, 47, 664, (2004) [11] Utsumi, T.; Kimura, H., Solutions of partial differential equations with the CIP-BS method, JSME int. J. B, 47, 761, (2004) [12] Ogata, Y.; Yabe, T., Multi-dimensional semi-Lagrangian characteristic approach to the shallow water equations by the CIP method, Int. J. comput. eng. sci., 5, 699, (2004) [13] Aoki, T., Interpolated differential operator (IDO) scheme for solving partial differential equations, Comput. phys. commun., 102, 132, (1997) [14] Aoki, T.; Nishita, S.; Sakurai, K., Interpolated differential operator scheme and application to level set method, Comput. fluid dynam. J., 9, 4, 418, (2001) [15] Aoki, T., 3D simulation for falling papers, Comput. phys. commun., 142, 326, (2001) · Zbl 1195.74275 [16] Kobara, T.; Aoki, T.; Tanahashi, M., Interpolated differential operator (IDO) scheme for direct numerical simulation of two-dimensional homogeneous isotropic turbulence, Trans. JSME B, 70, 2791, (2004), (in Japanese) [17] Chu, P.C.; Fan, C., A three-point combined compact difference scheme, J. comput. phys., 140, 370, (1998) · Zbl 0923.65071 [18] Harlow, F.H.; Welch, J.E., A numerical calculation of time dependent viscous incompressible flow of fluid with free surface, Phys. fluids, 8, 2182, (1965) · Zbl 1180.76043 [19] Amsden, A.A.; Harlow, F.H., A simplified MAC technique for incompressible fluid flow calculations, J. comput. phys., 6, 322, (1970) · Zbl 0206.55002 [20] Sakurai, K.; Aoki, T.; Lee, W.H.; Kato, K., Poisson equation solver with fourth-order accuracy by using interpolated differential operator scheme, Comput. math. appl., 43, 621, (2002) · Zbl 0999.65114 [21] Thompson, M.C.; Ferziger, J.H., An adaptive multigrid technique for the incompressible Navier-Stokes equations, J. comput. phys., 82, 94, (1989) · Zbl 0665.76034
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