Accelerating convergence for backward Euler and trapezoid time discretization schemes

Osman Raşit Işık; Tarkan Öner

Accelerating convergence for backward Euler and trapezoid time discretization schemes

Year 2014, Volume: 2 Issue: 3, 214 - 220, 01.12.2014

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.
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.

Osman Rasit Isik1and Tarkan ¨Oner2

Year 2014, Volume: 2 Issue: 3, 214 - 220, 01.12.2014

Osman Raşit Işık Tarkan Öner

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.

There are 7 citations in total.

Details

Journal Section	Articles
Authors	Osman Raşit Işık This is me Tarkan Öner This is me
Publication Date	December 1, 2014
Published in Issue	Year 2014 Volume: 2 Issue: 3

Cite

APA	Işık, O. R., & Öner, T. (2014). Accelerating convergence for backward Euler and trapezoid time discretization schemes. New Trends in Mathematical Sciences, 2(3), 214-220.
AMA	Işık OR, Öner T. Accelerating convergence for backward Euler and trapezoid time discretization schemes. New Trends in Mathematical Sciences. December 2014;2(3):214-220.
Chicago	Işık, Osman Raşit, and Tarkan Öner. “Accelerating Convergence for Backward Euler and Trapezoid Time Discretization Schemes”. New Trends in Mathematical Sciences 2, no. 3 (December 2014): 214-20.
EndNote	Işık OR, Öner T (December 1, 2014) Accelerating convergence for backward Euler and trapezoid time discretization schemes. New Trends in Mathematical Sciences 2 3 214–220.
IEEE	O. R. Işık and T. Öner, “Accelerating convergence for backward Euler and trapezoid time discretization schemes”, New Trends in Mathematical Sciences, vol. 2, no. 3, pp. 214–220, 2014.
ISNAD	Işık, Osman Raşit - Öner, Tarkan. “Accelerating Convergence for Backward Euler and Trapezoid Time Discretization Schemes”. New Trends in Mathematical Sciences 2/3 (December 2014), 214-220.
JAMA	Işık OR, Öner T. Accelerating convergence for backward Euler and trapezoid time discretization schemes. New Trends in Mathematical Sciences. 2014;2:214–220.
MLA	Işık, Osman Raşit and Tarkan Öner. “Accelerating Convergence for Backward Euler and Trapezoid Time Discretization Schemes”. New Trends in Mathematical Sciences, vol. 2, no. 3, 2014, pp. 214-20.
Vancouver	Işık OR, Öner T. Accelerating convergence for backward Euler and trapezoid time discretization schemes. New Trends in Mathematical Sciences. 2014;2(3):214-20.