A fifth-order combined compact difference scheme for Stokes flow on polar geometries. (English) Zbl 1426.76471
Summary: Incompressible flows with zero Reynolds number can be modeled by the Stokes equations. When numerically solving the Stokes flow in stream-vorticity formulation with high-order accuracy, it will be important to solve both the stream function and velocity components with the high-order accuracy simultaneously. In this work, we will develop a fifth-order spectral/combined compact difference (CCD) method for the Stokes equation in stream-vorticity formulation on the polar geometries, including a unit disk and an annular domain. We first use the truncated Fourier series to derive a coupled system of singular ordinary differential equations for the Fourier coefficients, then use a shifted grid to handle the coordinate singularity without pole condition. More importantly, a three-point CCD scheme is developed to solve the obtained system of differential equations. Numerical results are presented to show that the proposed spectral/CCD method can obtain all physical quantities in the Stokes flow, including the stream function and vorticity function as well as all velocity components, with fifth-order accuracy, which is much more accurate and efficient than low-order methods in the literature.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
Stokes flow; combined compact difference (CCD) scheme; truncated Fourier series; shifted grid; coordinate singularityReferences:
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