Abstract
The Interpolated Differential Operator (IDO) scheme has been developed for the numerical solution of the fluid motion equations, and allows to produce highly accurate results by introducing the spatial derivative of the physical value as an additional dependent variable. For incompressible flows, semi-implicit time integration is strongly affected by the Courant and diffusion number limitation. A high-order fully-implicit IDO scheme is presented, and the two-stage implicit Runge-Kutta time integration keeps over third-order accuracy. The application of the method to the direct numerical simulation of turbulence demonstrates that the proposed scheme retains a resolution comparable to that of spectral methods even for relatively large Courant numbers. The scheme is further applied to the Local Mesh Refinement (LMR) method, where the size of the time step is often restricted by the dimension of the smallest meshes. In the computation of the Karman vortex street problem, the implicit IDO scheme with LMR is shown to allow a conspicuous saving of computational resources.
Similar content being viewed by others
References
Aoki T (1997) Interpolated Differential Operator (IDO) scheme for solving partial differential equations. Comput Phys Commun 102:132–146
Aoki T (2001) 3D simulation for falling papers. Comput Phys Commun 142:326–329
Imai Y, Aoki T Accuracy study of the IDO scheme by Fourier analysis. J Comp Phys submitted
Berger MJ, Oliger J (1984) Adaptive mesh refinement for hyperbolic partial differential equations. J Comp Phys 53:484–512
Berger MJ, Colella P (1989): Local adaptive mesh refinement for shock hydrodynamics. J Comp Phys 82:64–84
Imai Y, Aoki T (2004) A numerical method on Cartesian grid for Image-based blood flow simulation. Trans JSME A70: 1232–1239. (in Japanese)
Fujiwara K, Aoki T, Ogawa H (2004) Higher-order AMR based on IDO scheme for fluid-structure interaction. CD-ROM WCCM VI in conjunction with APCOM’04 Beijing China
Butcher JC (1964) Implicit Runge–Kutta processes. Math Comp 15:50–64
Yabe T, Aoki T (1991) A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver. Comput Phys Commun 66:219–232
Yabe T, Xiao F, Utsumi T (2001) The constrained interpolation profile method for multiphase analysis. J Comp Phys 169:556–593
Aoki T, Nishita S, Sakurai K (2001) Interpolated differential operator scheme and application to level set method. Comput Fluid Dyn J 9(4):418–426
Gottlieb D, Orszag A (1977) Numerical analysis of spectral methods. SIAM, Philadelphia
Amsden AA, Harlow FH (1970) A simplified MAC technique for incompressible fluid flow calculations. J Comp Phys 6:322–325
Sakurai K, Aoki T, Lee WH, Kato K (2002) Poisson equation solver with fourth-order accuracy by using interpolated differential operator scheme. Comput Math Appl 43:621–630
Thompson MC, Ferziger JH (1989) An adaptive multigrid technique for the incompressible Navier-Stokes equations. J Comp Phys 82:94–121
Imai Y, Aoki T Stable coupling between vector and scalar variables for the IDO scheme on a collocated grid. J Comp Phys submitted
Tezduyar T, Aliabadi S, Behr M (1998) Enhanced-Discretization Interface-Capturing Technique (EDICT) for computation of unsteady flows with interfaces. Comput Methods Appl Mech Eng 155:235–248
Tezduyar T, Osawa Y (2001) The multi-domain method for computation of the aerodynamics of a parachute crossing the far wake of an aircraft. Comput Methods Appl Mech Eng 191:705–716
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Imai, Y., Aoki, T. A higher-order implicit IDO scheme and its CFD application to local mesh refinement method. Comput Mech 38, 211–221 (2006). https://doi.org/10.1007/s00466-005-0742-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-005-0742-x