Numerical contour integral methods for unsteady Stokes equations

The unsteady Stokes equations are semi-discretized in space to obtain a system of linear time-invariant differential-algebraic equations (DAEs), i.e., the unsteady discrete Stokes equations. The solution to unsteady discrete Stokes equations is represented as an integral along a smooth curve Γ in the complex plane with singularities of the integrand located on the left of and not too close to the curve Γ. Truncated quadrature rules based on the sinc function are then employed to evaluate the solution. This results in a number of complex linear systems to solve, leading to major expense in practical implementation. Constraint preconditioners are proposed to work with the Krylov subspace methods for solving those complex linear systems. Numerical examples illustrate that the numerical contour integral methods are more effective than the time-stepping methods. In addition, the constraint preconditioners significantly improve the behavior of Krylov subspace methods for solving the involved complex linear systems.