The immersed boundary (IB) method, introduced by Charles S. Peskin, is a mathematical formulation and numerical scheme for simulating fluid-structure interaction, particularly in biological fluid dynamics. The method combines Eulerian and Lagrangian variables, linked by the Dirac delta function, to model the interaction between fluids and elastic structures. Eulerian variables are defined on a fixed Cartesian mesh, while Lagrangian variables are defined on a moving curvilinear mesh. The interaction equations involve a smoothed approximation of the Dirac delta function, and the transfer of data between meshes is governed by Eulerian/Lagrangian identities. Temporal discretization is achieved using a second-order Runge–Kutta method. The IB method is versatile, applicable to a wide range of problems, including fluid-structure interaction, and has been used to study phenomena such as heart valve dynamics. The method's mathematical structure is derived from the principle of least action, and its numerical implementation involves spatial discretization on two grids: one for Eulerian variables and one for Lagrangian variables. The method ensures conservation of mass, momentum, and energy, and its effectiveness is demonstrated through various applications in fluid dynamics. The IB method's ability to handle complex interactions between fluids and structures makes it a powerful tool in computational fluid dynamics.The immersed boundary (IB) method, introduced by Charles S. Peskin, is a mathematical formulation and numerical scheme for simulating fluid-structure interaction, particularly in biological fluid dynamics. The method combines Eulerian and Lagrangian variables, linked by the Dirac delta function, to model the interaction between fluids and elastic structures. Eulerian variables are defined on a fixed Cartesian mesh, while Lagrangian variables are defined on a moving curvilinear mesh. The interaction equations involve a smoothed approximation of the Dirac delta function, and the transfer of data between meshes is governed by Eulerian/Lagrangian identities. Temporal discretization is achieved using a second-order Runge–Kutta method. The IB method is versatile, applicable to a wide range of problems, including fluid-structure interaction, and has been used to study phenomena such as heart valve dynamics. The method's mathematical structure is derived from the principle of least action, and its numerical implementation involves spatial discretization on two grids: one for Eulerian variables and one for Lagrangian variables. The method ensures conservation of mass, momentum, and energy, and its effectiveness is demonstrated through various applications in fluid dynamics. The IB method's ability to handle complex interactions between fluids and structures makes it a powerful tool in computational fluid dynamics.