Abstract
AbstractDahlquist, Liniger, and Nevanlinna design a family of one-leg, two-step methods (the DLN method) that is second order, $$\varvec{A}-$$
A
-
and $$\varvec{G}-$$
G
-
stable for arbitrary, non-uniform time steps. Recently, the implementation of the DLN method can be simplified by the refactorization process (adding time filters on the backward Euler scheme). Due to these fine properties, the DLN method has strong potential for the numerical simulation of time-dependent fluid models. In the report, we propose a semi-implicit DLN algorithm for the Navier-Stokes equations (avoiding non-linear solver at each time step) and prove the unconditional, long-term stability, and second-order convergence with the moderate time step restriction. Moreover, the adaptive DLN algorithms by the required error or numerical dissipation criterion are presented to balance the accuracy and computational cost. Numerical tests will be given to support the main conclusions.
Publisher
Springer Science and Business Media LLC
Reference55 articles.
1. Baker, G.A.: Galerkin approximation for the Navier-Stokes equations. Technical report, Harvard University, (1976)
2. Baker, G.A., Dougalis, V.A., Karakashian, O.A.: On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comp. 39(160), 339–375 (1982)
3. Bertagna, L., Quaini, A., Veneziani, A.: Deconvolution-based nonlinear filtering for incompressible flows at moderately large Reynolds numbers. Internat. J. Numer. Methods Fluids 81(8), 463–488 (2016)
4. Bukač, M., Seboldt, A., Trenchea, C.: Refactorization of Cauchy’s method: a second-order partitioned method for fluid-thick structure interaction problems. J. Math. Fluid Mech. 23(3):Paper No. 64, 25, (2021)
5. Bukač, M., Trenchea, C.: Adaptive, second-order, unconditionally stable partitioned method for fluid-structure interaction. Comput. Methods Appl. Mech. Engrg. 393(14847), 24 (2022)