Numerical implementation of an iterative method with boundary condition splitting for solving the nonstationary stokes problem on the basis of an asymptotically stable two-stage difference scheme

A new numerical implementation of a fast-converging iterative method with splitting of boundary conditions is constructed for solving the Dirichlet initial-boundary value problem for the nonstationary Stokes system. The method was earlier proposed and substantiated at the differential level by B.V. Pal’tsev. The problem is considered in a strip and is assumed to be periodic along the strip. According to the numerical implementation proposed, a special vector parabolic problem for velocity approximations (which arises at iterations of the method) is discretized using an asymptotically stable two-stage difference scheme that is second-order accurate in time. The spatial discretization is based on bilinear finite elements on uniform rectangular grids. A numerical study shows that the convergence rate of the constructed iterative method is as high as that of the original method at the differential level (the error is reduced by approximately 7 times per iteration step). For velocities, the method is second-order accurate in the max norm. For pressures, the method is second-order accurate in space and first-order accurate in time.