Spurious transients of projection methods in microflow simulations

Abstract

The temporal behavior of projection methods for viscous incompressible low-Reynolds-number flows is addressed. The methods considered result from algebraically splitting the linear system corresponding to each time step, in such a way that the computation of velocity is segregated from that of pressure. Each method is characterized by two (possibly equal) approximate inverses (B1 and B2) of the momentum-equation velocity matrix, plus a parameter γ which renders the method non-incremental (if γ=0) or incremental (if γ=1). The classical first-order projection method, together with more sophisticated methods (Perot’s second-order method, Yosida method, pseudo-exact factorization method) and their incremental variants are put into the same algebraic form and their accuracy numerically tested. Splitting errors of first, second and third order in the time step size δt are obtained, depending on the method. The methods are then discussed in terms of their ability and efficiency to compute steady states. Non-incremental methods are impractical because extremely small time steps are required for the steady state, which depends on δt, to be reasonably accurate. Incremental methods, on the other hand, either become unstable as δt is increased or develop a remarkable spurious transient which may last an extremely long time (much longer than any physical time scale involved). These transients have serious practical consequences on the simulation of steady (or slowly varying), low-inertia flows. From the physical viewpoint, the spurious transients may interfere with true slow processes of the system, such as heat transfer or species transport, without showing any obvious symptoms (wiggly behavior in space or time, for example, do not occur). From the computational viewpoint, the limitation in time step imposed by the spurious transient phenomenon weighs against choosing projection schemes for microflow applications, despite the low cost of each time step.