Skew-Symmetric MFS and Harmonic-Polynomial-Type Methods to Solve the Three-Dimensional Stokes Equations

Source abstract

Based on a new general solution, we derive three skew-symmetric variants of the method of fundamental solutions (MFS) for solving the three-dimensional Stokes equations: the symmetric/skew-symmetric MFS (SSMFS), the separated MFS (SeMFS) as an extension of the conventional MFS that incorporates an additional skew-symmetric part, and the symmetric/skew-symmetric separated MFS (SSeMFS) as an extension of the SSMFS by further adding a skew-symmetric component. In these methods, a skew-symmetric tensor interlaced with the Oseen tensor is employed to regularize the Stokeslet. Numerical examinations show that the SeMFS and SSeMFS outperform the conventional MFS, achieving faster convergence and an improvement in accuracy of approximately one order of magnitude. We construct six families of kth-order homogeneous harmonic polynomials that possess rich properties. These polynomials are arranged into two harmonic vectors, which are then inserted into the Liu–Hsu–Tsai formula to yield the LHT method (LHTM), into the Liu–Tsai formula to yield the LT method (LTM), and into the Slabodyanskii formula to yield the Slabodyanskii method (SM). The resulting methods solve the Stokes equations via a simple collocation technique that enforces the prescribed boundary conditions. In addition, we derive a particular solution of the Stokes equations by means of an integration method (IM). Six numerical examples, including a benchmark three-dimensional lid-driven cavity problem, are tested to assess the efficiency and accuracy of these newly developed harmonic-polynomial-type methods. They are superior to the MFS-type methods, delivering improvements in accuracy of several orders of magnitude together with much faster convergence rates.