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Analysis of a one-dimensional model for the immersed boundary method. (English) Zbl 0762.65052
The accuracy of C. S. Peskin’s immersed boundary method [J. Comput. Phys. 25, 220-252 (1977; Zbl 0403.76100)] is analyzed for one-dimensional model problems. Differential equations of the form \(u_ t=u_{xx}+c(t)\delta(x-\alpha(t))\) are considered. The delta function \(\delta(x)\) is replaced by its discrete approximation and the obtained equation is solved by a Crank-Nicolson method on a uniform grid.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76Z05 Physiological flows
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