Initial-Boundary-Value Problems for the Stokes and Navier–Stokes Equations on Staggered Grids

Sammanfattning: In the first part of the thesis various types of boundary conditions for the steady state Stokes equations are considered. We formulate the boundary conditions in a new way, such that the boundary value problem becomes non-singular, and derive estimates of the solution. By using a second-order finite difference approximation on a staggered grid, we are able to derive a non-singular approximation in a direct way. Furthermore, we derive the same type of estimates as for the continuous case.The estimates of the solution are derived for the case where the velocity field is prescribed at the boundary, and moreover for the case where the pressure is specified instead of one of the velocity components. We also show that the same estimates hold when the continuity condition is substituted by the Poisson equation for the pressure if we add the divergence condition as a boundary condition.Next, by using a Laplace transform in time for the analysis, we are able to derive the same type of estimates for the linearized Navier-Stokes equations as for the steady state Stokes equations. Also here, we make an analogous analysis for a difference approximation on a staggered grid, and derive the corresponding estimates.Numerical experiments confirm the theoretical results. We also show that the results are valid for more complicated geometries, by using composite overlapping grids, with local velocity components on orthogonal staggered subgrids.The analysis and derivation of the estimates here, give new insight into the behavior of the solutions and are used for developing a fourth-order method for the incompressible Navier-Stokes equations. In order to eliminate problems with non-physical parasitic solutions and extra numerical boundary conditions, we construct a finite difference scheme with compact high order difference approximations of Padé type on a staggered grid. We use the same type of boundary conditions proposed, developed and analyzed for the linearized Navier-Stokes equations. Several numerical experiments demonstrate the accuracy and efficiency of the method.