Immersed Boundary Method for Variable Viscosity and Variable Density Problems Using Fast Constant-Coefficient Linear Solvers II: Theory

We analyze the stability and convergence of first-order accurate and second-order accurate timestepping schemes for the Navier-Stokes equations with variable viscosity. These schemes are characterized by a mixed implicit/explicit treatment of the viscous term, in which a numerical parameter, λ, determines the degree of splitting between the implicit and explicit contributions. The reason for this splitting is that it avoids the need to solve computationally expensive linear systems that may change at each time step. Provided the parameter λ is within a permissible range, we prove that the first-order accurate and second-order accurate schemes are convergent. We show further that the efficiency of the second-order accurate scheme depends on how λ is chosen within the permissible range, and we discuss choices that work well in practice. We use parameters motivated by this analysis to simulate internal gravity waves, which arise in stratified fluids with variable density. We examine how the wave properties change in the nonlinear and variable viscosity regime, and we use these simulations to test how well our theory predicts the efficiency of the second-order accurate timestepping scheme in practice.