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Implicit time accurate simulation of unsteady flow
Author(s) -
van Buuren René,
Kuerten Hans,
Geurts Bernard J.
Publication year - 2001
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/1097-0363(20010330)35:6<687::aid-fld110>3.0.co;2-q
Subject(s) - discretization , mathematics , convergence (economics) , solver , stability (learning theory) , flow (mathematics) , context (archaeology) , numerical integration , computer science , mathematical optimization , mathematical analysis , geometry , paleontology , machine learning , economics , biology , economic growth
Implicit time integration was studied in the context of unsteady shock‐boundary layer interaction flow. With an explicit second‐order Runge–Kutta scheme, a reference solution to compare with the implicit second‐order Crank–Nicolson scheme was determined. The time step in the explicit scheme is restricted by both temporal accuracy as well as stability requirements, whereas in the A‐stable implicit scheme, the time step has to obey temporal resolution requirements and numerical convergence conditions. The non‐linear discrete equations for each time step are solved iteratively by adding a pseudo‐time derivative. The quasi‐Newton approach is adopted and the linear systems that arise are approximately solved with a symmetric block Gauss–Seidel solver. As a guiding principle for properly setting numerical time integration parameters that yield an efficient time accurate capturing of the solution, the global error caused by the temporal integration is compared with the error resulting from the spatial discretization. Focus is on the sensitivity of properties of the solution in relation to the time step. Numerical simulations show that the time step needed for acceptable accuracy can be considerably larger than the explicit stability time step; typical ratios range from 20 to 80. At large time steps, convergence problems that are closely related to a highly complex structure of the basins of attraction of the iterative method may occur. Copyright © 2001 John Wiley & Sons, Ltd.