Over-determined formulation of the immersed boundary conditions method

The over-determined formulation of the immersed boundary conditions (IBC) method is proposed. The method relies on the Fourier expansions in the periodic direction and Chebyshev expansions in the transverse direction. The boundaries of the physical domain are immersed inside a regular computational domain and the boundary conditions enter the algorithm in the form of constraints. Construction of these constraints provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. Use of the additional degrees of freedom that leads to an over-determined system is explored in order to improve the accuracy of the IBC method and to expand its applicability to more extreme geometries. The over-constraint formulation has been tested on three model problems that lead to the Laplace, biharmonic and Navier–Stokes equations and thus cover the most commonly encountered types of operators. In all cases tested the over-determined formulation was found to improve the performance of the IBC method.