Numerical simulation of fluid flow with a fast Boundary-Domain Integral Method

For the simulation of fluid flow the velocity-vorticity formulation of the Navier-Stokes equations can be used. Approximating the first order time derivative in these equations by finite differences, essentially, the Yukawa equation for the velocity has to be solved with an inhomogeneity governed by the vorticity. Applying the Boundary Element Method to such problems is not straightforward. However, the Boundary-Domain Integral Method (BDIM) can be used, which requires to solve beside the boundary integrals as well a domain integral due to the inhomogeneity. Classical BEM formulations have a quadratic complexity with respect to the unknowns on the boundary and, for the BDIM additionally, for the unknowns in the domain. This prevents the BDIM from being applied to real-world problems.

To reduce the quadratic complexity in BEM, approximation methods like the fast multipole method, ℋ-matrices in combination with the adaptive cross approximation or the ℋ2 -matrix technique can be used. All these methods reduce the complexity to an almost linear order. Here, the ℋ2 -matrix technique is applied to reduce storage and computing time for the right hand side, i.e., for the matrix of the domain integrals. Based on interpolatory kernel expansions and an heuristic criterium for too small matrix entries an almost linear complexity can be achieved. The numerical studies focus on a lid-driven cavity. The complexity reduction is demonstrated.