Application of the Fictitious Domain Method for Navier-Stokes Equations

To apply the fictitious domain method and conduct numericalexperiments, a boundary value problem for an ordinary differential equation is considered. The results of numerical calculations for different valuesof the iterative parameter τ and the small parameter ε are presented. Astudy of the auxiliary problem of the fictitious domain method for NavierStokes equations with continuation into a fictitious subdomain by highercoefficients with a small parameter is carried out. A generalized solutionof the auxiliary problem of the fictitious domain method with continuationby higher coefficients with a small parameter is determined. After all theabove mathematical studies, a computational algorithm has been developedfor the numerical solution of the problem. Two methods were used to solvethe problem numerically. The first variant is the fictitious domain methodassociated with the modification of nonlinear terms in a fictitious subdomain.The model problem shows the effectiveness of using such a modification. Theproposed version of the method is used to solve two problems at once that arisewhile numerically solving systems of Navier-Stokes equations: the problem ofa curved boundary of an arbitrary domain and the problem of absence of aboundary condition for pressure in physical formulation of the internal flowproblem. The main advantage of this method is its universality in developmentof computer programs. The second method used calculation on a uniform gridinside the area. When numerically implementing the solution on a uniformgrid inside the domain, using this method it’s possible to accurately take intoaccount the boundaries of the curved domain and ensure the accuracy of thevalue of the function at the boundaries of the domain. Methodical calculationswere carried out, the results of numerical calculations were obtained. Whenconducting numerical experiments in both cases, quantitative and qualitativeindicators of numerical results coincide.