Abstract
Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations,
In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, SIAM J Sci Comput 31(2):1035–1062, 2008), with other short-recurrences Krylov methods, specifically the Bi-Conjugate Gradient Method (Bi-CG) (Fletcher, Conjugate gradient methods for indefinite systems. In: Proceedings of the Dundee conference on numerical analysis, pp 73–89, 1976), restarted Generalized Minimal Residual (GMRES(m)) (Saad and Schultz, SIAM J Sci Stat Comput 7:856–869, 1986), and Bi-Conjugate Gradient Stabilized method (Bi-CGSTAB) (van der Vorst, SIAM J Sci Stat Comput 13(2):631–644, 1992).
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References
R. Fletcher, Conjugate gradient methods for indefinite systems, ed. by G.A. Watson. Proceedings of the Dundee Biennal Conference on Numerical Analysis 1974 (Springer, New York, 1975), pp. 73–89
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edn. (Society for Industrial and Applied Mathematics, Philadelphia, 2003)
Y. Saad, M. Schultz, GMRES: a generalized minimal residual algorithm for solving nonsymetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
G.L.G. Sleijpen, D.R. Fokkema, BiCGstab(ℓ) for linear equations involving Unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Anal. 1, 11–32 (1993)
G.L.G. Sleijpen, P. Sonneveld, M.B. van Gijzen, Bi-CGSTAB as an induced dimension reduction method. Appl. Numer. Math. 60, 1100–1114 (2010)
V. Simoncini, D.B. Szyld, Interpreting IDR as a Petrov-Galerkin method. SIAM J. Sci. Comput. 32 (4), 1898–1912 (2010)
G.L.G. Sleijpen, H.A. van der Vorst, Maintaining convergence properties of bicgstab methods in finite precision arithmetic. Numer. Algorithms 10 (2), 203–223 (1995)
G.L.G. Sleijpen, M.B. van Gijzen, Exploiting BiCGstab(ℓ) strategies to induce dimension reduction. SIAM J. Sci. Comput. 32 (5), 2687–2709 (2010)
P. Sonneveld, M.B. van Gijzen, IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations. SIAM J. Sci. Comput. 31 (2), 1035–1062 (2008)
H.A. van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (2), 631–644 (1992)
M.B. van Gijzen, P. Sonneveld, Algorithm 913: an elegant IDR(s) variant that efficiently exploits bi-orthogonality properties. ACM Trans. Math. Softw. 38 (1), 5:1–5:19 (2011)
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Astudillo, R., van Gijzen, M.B. (2016). The Induced Dimension Reduction Method Applied to Convection-Diffusion-Reaction Problems. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_29
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