Abstract
In this study, we introduce two algorithms to numerically solve any initial value problem (IVP). These algorithms dependon time relaxation model (TRM) which is obtained adding a time relaxation term into IVP. Discretizing TRM by using backward Euler(BE) method gives the first algorithm. Similarly, the second algorithm is followed by using trapezoid (TR) time stepping scheme . Undersome conditions, the first algorithm increases the order of convergence from one to two and the second one increases the order fromtwo to three. Thus, more accurate results can be obtained. To verify the accuracy of the methods, they are applied to some numericalexamples. Numerical results overlap with the theoretical results
Keywords
Relaxation model backward Euler method accelerating convergence trapezoid time discretization
References
- N.A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in LES, Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, 2001.
- V.J. Ervin, W. Layton and M. Neda, Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Mod., 4 (2007) 648–670.
- O.R. Isik, Spin up problem and accelerating convergence to steady state, Appl. Math. Modell. (2013) 3242-3253.
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- W. Layton, C. David Pruett and L. G. Rebholz, Temporally regularized direct numerical simulation, Appl. Math. Comp. 216 (2010) 3728–3738.
- M. Neda, Discontinuous Time Relaxation Method for the Time-Dependent Navier-Stokes Equations, Adv. Numer. Anal., Volume 2010, doi:10.1155/2010/419021.
- C.D. Pruett, T.B. Gatski, C.E. Grosch and W.D. Thacker, The temporally filtered Navier–Stokes equations: properties of the residual- stress, Phys. Fluids, 15 (2003) 2127–2140.
Abstract
References
- N.A. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in LES, Modern Simulation Strategies for Turbulent Flow, R. T. Edwards, 2001.
- V.J. Ervin, W. Layton and M. Neda, Numerical analysis of a higher order time relaxation model of fluids, Int. J. Numer. Anal. Mod., 4 (2007) 648–670.
- O.R. Isik, Spin up problem and accelerating convergence to steady state, Appl. Math. Modell. (2013) 3242-3253.
- W. Layton and M. Neda, Truncation of scales by time relaxation, J. Math. Anal. Appl,325 ( 2007) 788–807.
- W. Layton, C. David Pruett and L. G. Rebholz, Temporally regularized direct numerical simulation, Appl. Math. Comp. 216 (2010) 3728–3738.
- M. Neda, Discontinuous Time Relaxation Method for the Time-Dependent Navier-Stokes Equations, Adv. Numer. Anal., Volume 2010, doi:10.1155/2010/419021.
- C.D. Pruett, T.B. Gatski, C.E. Grosch and W.D. Thacker, The temporally filtered Navier–Stokes equations: properties of the residual- stress, Phys. Fluids, 15 (2003) 2127–2140.
Details
| Journal Section | Articles |
|---|---|
| Authors | |
| Publication Date | December 1, 2014 |
| Published in Issue | Year 2014 Volume: 2 Issue: 3 |