Geodesic
Iterative approach to solving boundary integral equations
Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 4, pp. 54-67

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Under consideration are the issues of numerical solution of a boundary integral equation describing the vorticity generation process on the streamlined airfoils in meshless vortex methods. The traditional approach based on the quadrature method leads to the necessity of solving a system of linear algebraic equations with dense matrix. If we consider the system of airfoils moving relative to one another, this procedure has to be performed at each time step of the calculation, and its high computational complexity significantly reduces the efficiency of vortex methods. The transition from the traditional approach expressed by an integral equation of the first kind to an approach with the integral equation of the second kind makes it possible to apply the simple-iteration method for numerical solving the boundary integral equation. By examples of some model problems, we demonstrate that the iterative approach allows reducing the computational complexity of the problem by tens to hundreds times while providing an acceptable accuracy of the approximate solution.
Keywords: vortex method, incompressible flow, vortex sheet, boundary integral equation, singular integral, simple-iteration method. 
@article{SJIM_2019_22_4_a5,
     author = {E. A. Mikhailov and I. K. Marchevskii and K. S. Kuzmina},
     title = {Iterative approach to solving boundary integral equations},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {54--67},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/}
}
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E. A. Mikhailov; I. K. Marchevskii; K. S. Kuzmina. Iterative approach to solving boundary integral equations. Sibirskij žurnal industrialʹnoj matematiki, Tome 22 (2019) no. 4, pp. 54-67. http://geodesic.mathdoc.fr/item/SJIM_2019_22_4_a5/