Non-relaxed finite volume fractional step schemes for unsteady incompressible flows

Despite their well-established efficiency and accuracy, fractional-step schemes are not commonly used in finite volume methods. This article presents first-, second-, and third-order in-time fractional step schemes to solve incompressible convective time-dependent flows using collocated meshes. The fractional step methods are designed from a fully discrete problem and allow for optimal convergence rates. A detailed algebraic derivation of the method is included, incorporating implementation details to easily adapt existing finite volume codes. The schemes use extrapolations for pressure in the first calculation step and include Yosida's approximation in the last step to ensure third-order in-time accuracy. In addition, an incomplete Yosida scheme that considers a correction only in the diffusive term is evaluated. Time discretization is achieved using the backward differentiation formula, including first-, second-, and third-order in-time schemes, and an optimized second-order one. The linear momentum and continuity equations are decoupled using a pressure-correction strategy. The numerical results include a converged test using the solutions to evaluate the convergence in-time rates. Finally, the dominant convective flows related to the lid-driven cavity problem are solved for two-and three-dimensional flows. The use of Picard, Newton, and Picard-Newton linearization is analyzed. The optimal rates of the convergence errors of the fractional-step methods are verified. The results indicate that Yosida's methods are more accurate than the projection method for the same meshes and time-step sizes. The use of Newton and Picard-Newton linearization strategies considerably reduces the number of iterations compared with the Picard scheme. The optimized second-order method is more accurate than the classical one and is comparable to the third-order method in the solution of dominant convective flows. The benefits of using high-order time integration schemes are verified.

Automated summary

This article introduces fractional-step schemes for solving incompressible convective time-dependent flows. The schemes are designed from a fully discrete problem and allow for optimal convergence rates. Results showed that Yosida's methods are more accurate than the projection method, and the use of Newton and Picard-Newton linearization strategies considerably reduces the number of iterations compared with the Picard scheme. The optimized second-order method is more accurate than the classical one and is comparable to the third-order method in the solution of dominant convective flows. The benefits of using high-order time integration schemes are verified.