MFS with time-dependent fundamental solutions for unsteady Stokes equations.

*(English)*Zbl 1195.76324Summary: This paper describes the applications of the method of fundamental solutions (MFS) for 2D and 3D unsteady Stokes equations. The desired solutions are represented by a series of unsteady Stokeslets, which are the time-dependent fundamental solutions of the unsteady Stokes equations. To obtain the unknown intensities of the fundamental solutions, the source points are properly located in the time-space domain and then the initial and boundary conditions at the time-space field points are collocated. In the time-marching process, the prescribed collocation procedure is applied in a time-space box with suitable time increment, thus the solutions are advanced in time. Numerical experiments of unsteady Stokes problems in 2D and 3D peanut-shaped domains with unsteady analytical solutions are carried out and the effects of time increments and source points on the solution accuracy are studied. The time evolution of history of numerical results shows good agreement with the analytical solutions, so it demonstrates that the proposed meshless numerical method with the concept of space-time unification is a promising meshless numerical scheme to solve the unsteady Stokes equations. In the spirit of the method of fundamental solutions, the present meshless method is free from numerical integrations as well as singularities in the spatial variables.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

65M80 | Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs |

##### Keywords:

method of fundamental solutions; unsteady Stokes equations; unsteady stokeslets; multi-dimensions
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\textit{C. C. Tsai} et al., Eng. Anal. Bound. Elem. 30, No. 10, 897--908 (2006; Zbl 1195.76324)

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##### References:

[1] | Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—I: surface approximations and partial derivative estimates, Comput math appl, 19, 127-145, (1990) · Zbl 0692.76003 |

[2] | Kansa, E.J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—II: solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput math appl, 19, 147-161, (1990) · Zbl 0850.76048 |

[3] | Kansa, E.J.; Power, H.; Fasshauer, G.E.; Ling, L., A volumetric integral radial basis function method for time-dependent partial differential equations—I: formulation, Eng anal bound elem, 28, 1191-1206, (2004) · Zbl 1159.76363 |

[4] | Ling, L.; Opfer, R.; Schaback, R., Results on meshless collocation techniques, engng, Anal bound elem, 30, 247-253, (2006) · Zbl 1195.65177 |

[5] | Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR comput math math phys, 4, 82-126, (1964) · Zbl 0154.17604 |

[6] | Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J num anal, 14, 638-650, (1977) · Zbl 0368.65058 |

[7] | Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J num anal, 22, 644-669, (1985) · Zbl 0579.65121 |

[8] | Trefftz E. Ein Gegenstuck zum ritzschen Verfahren. In: Proceedings of the second international congress of applied mechanics. 1926, Zurich. p. 131-37. |

[9] | Kita, E.; Kamiya, N., Trefftz method: an overview, Adv eng software, 24, 3-12, (1995) · Zbl 0984.65502 |

[10] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074 |

[11] | Chen, C.S.; Golberg, M.A.; Hon, Y.C., The method of fundamental solutions and quasi-Monte-Carlo method for diffusion equations, Int J num meth eng, 43, 1421-1435, (1998) · Zbl 0929.76098 |

[12] | Tsai CC. Meshless Numerical Methods and their Engineering Applications. PhD thesis, Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, 2002. |

[13] | Young, D.L.; Tsai, C.C.; Fan, C.M., Direct approach to solve non-homogeneous diffusion problems using fundamental solutions and dual reciprocity methods, J chin inst eng, 27, 4, 597-609, (2004) |

[14] | Young, D.L.; Tsai, C.C.; Murugesan, K.; Fan, C.M.; Chen, C.W., Time-dependent fundamental solutions for homogeneous diffusion problems, Eng anal bound elem, 28, 1463-1473, (2004) · Zbl 1098.76622 |

[15] | Ho, C.M.; Tai, Y.C., Micro-electro-mechanical-system (MEMS) and fluid flows, Ann rev fluid mech, 12, 579-612, (1998) |

[16] | Tsai, C.C.; Young, D.L.; Cheng, A.H.D., Meshless BEM for three-dimensional Stokes flows, CMES: comput modeling eng sci, 3, 117-128, (2002) · Zbl 1147.76595 |

[17] | Young, D.L.; Chen, C.W.; Fan, C.M.; Murugesan, K.; Tsai, C.C., Method of fundamental solutions for Stokes flows in a rectangular cavity with cylinders, Eur J mech B/fluid, 24, 703-716, (2005) · Zbl 1103.76319 |

[18] | Alves, C.J.S.; Silvestre, A.L., Density results using stokeslets and a method of fundamental solution for the Stokes equations, Eng anal bound elem, 28, 1245-1252, (2004) · Zbl 1079.76058 |

[19] | Banerjee, P.K., The boundary element methods in engineering, (1994), McGraw-Hill New York |

[20] | Golberg, M.A.; Chen, C.S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (), 103-176 · Zbl 0945.65130 |

[21] | Chen, C.S.; Golberg, M.A.; Schaback, R.S., Recent developments of the dual reciprocity method using compactly supported radial basis functions, () · Zbl 1043.65133 |

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