HYBRID FDM-STABILIZED LANCZOS-TYPE IN SOLVING PDE PROBLEMS
Article 2
DOI:
https://doi.org/10.46754/jmsi.2021.12.002Keywords:
Finite Difference, Lanczos solvers, PDEs, SLEsAbstract
This study investigates the combination of finite difference method (FDM) and the stabilized Lanczos method to solve various partial differential equation (PDE) problems. This combination is wrapped in the algorithms called hybrid FDMRMEIEMLA and hybrid FDM-RLMinRes. FDM is the discretization method which converts the PDEs into algebraic formula, whereas both RMEIEMLA and RLMinRes are known as the stabilized Lanczos methods in solving largescale problems of SLEs. Their hybrids enable us to find the solutions of PDE problems accurately. There are at least three types of PDEs solved in this study, namely Helmholtz, wave, and heat equations. The convergence rate of our methods computed using the residual norms || b - Axk ||. Numerical results showed that our proposed methods performed well in solving the various PDEs with small residual norms.
References
T. Hillen. (2014). Partial Differential Equations: Theory and completely solved problems. Canada: John Wiley & Sons.
R. L. Burden, & J. D. Faires. (2010). Numerical Analysis (9th ed.). Canada: Brooks/Cole.
C. Lanczos. (1950). An iteration method for the solution of the eigenvalue problem of the linear differential and integral operators.
J. Res. Nat. Bur. Stand., 45(4), 255-282. https://doi.org/10.6028/jres.045.026.
C. Lanczos. (1952). Solution of system of linear equations by minimized iterations. J. Res. Nat. Bur. Stand., 49(1), 33-53. https://doi.org/10.6028/jres.049.006.
C. Brezinski, & H. Sadok. (1993). Lanczostype algorithms for solving systems of linear equations. Journal of Applied Numerical Mathematics, 11(6), 443-473. https://doi.org/10.1016/0168-9274(93)90087-8.
C. Baheux. (1995). New implementation of Lanczos method. Journal of Computational and Applied Mathematics, 57(1-2), 3-15. https://doi.org/10.1016/0377-0427(93) E0230-J.
M. Farooq, & A. Salhi. (2012). New recurrence relationships between orthogonal polynomials which lead to new Lanczostype algorithms. Journal of Prime Research in Mathematics, 8, 61-75.
C. Brezinski, & M. R. Zaglia. (1994). Breakdowns in the computation of orthogonal polynomials. Nonlinear Numerical Methods and Rational Approximation II, 296, 49–59. https://doi. org/10.1007/978-94-011-0970-3_5.
C. Brezinski, & M. R. Zaglia. (1995). Lookahead in Bi-CGSTAB and other methods for linear systems. BIT, 35(2), 169–201. http://dx.doi.org/10.1007/BF01737161.
A. B. Maharani, & A. Salhi. (2015). Restarting from specific points to cure breakdown in Lanczos-type algorithms. Journal of Mathematical and & Fundamental Science, 47(2), 167-184. http://dx.doi.org/10.5614/j.math.fund.sci.2015.47.2.5.
A. B. Maharani, A. Salhi, & R. A. Suharto. (2018). Introduction of interpolation and extrapolation model in Lanczos-type algorithms A13/B6 and A13/B3 to enhance their stability. Journal of Mathematical and Fundamental Science, 50(2), 148-165. https://doi.org/10.5614/j.math.fund.sci.2018.50.2.4.
A. B. Maharani, & A. Salhi. (2019). RMEIEMLA: The recent advance in improving the robustness of Lanczostype algorithms. AIP Conference Proceedings, 2138(1). http://dx.doi.org/10.1063/1.5121046.
A. B. Maharani, A. Juliansyah, R. Thalib, & D. A. Pratama. (2021). High performances of stabilized Lanczos-Types for solving high dimension problems: A survey. Int. J. of Math. and Comp. Sci., 16(2), 837-854. https://doi.org/10.5281/zenodo.4431070.
V. Grigoryan. (2010). Partial Differential Equations. Lecture note, University of California, Santa Barbara, version 0.1. Accessed April 8, 2021, from https://web.math.ucsb.edu/~grigoryan/124A.pdf
S. Thirumugam. (2012). Numerical methods for hyperbolic Partial Differential Equation, Department of Physics School of Physical,
Chemical and Applied Sciences Pondicherry University. Accessed April 5, 2021 from https://www.academia.edu/1827576/Numerical_Methods_for_Hyperbolic_PDE
L. Zhang. (2002). The finite difference method for Helmholtz equation with application of cloaking. Accessed April 5, 2021, from
https://math.missouristate.edu/assets/Math/li.pdf
M. Witten. (2014). Hyperbolic Partial Differential Equations: Populations, reactors, tides and waves: Theory and applications. Chicago: Elsevier.
B. E. Rapp. (2017). Microfluidics: Modelling, mechanics and mathematics. Elsevier.
A. Z. Azmi. (2007). Numerical solution for heat equation, University Technology Malaysia. Accessed April 8, 2021, from
https://www.nepjol.info/index.php/JNPhysSoc/article/view/34858/27355
A. Schikorra. (2017). Parabolic Partial Differential Equations; lecture notes [PDF]. http://www.pitt.edu/~armin/PDEIIWS16/pde2.p
L. Niragire. (2013). 2D numerical heat inverse model for diffusivity coefficients in cartesian coordinates. (master thesis, Indiana University of Pennsylvania). Accessed April 5, 2021, from https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.881.3549&rep=rep1&type=pdf
S. K. Njogu. (2009). Heat equations and their applications (one and two dimension heat equations). (master thesis, University of Nairobi). Accessed April 5, 2021, from http://erepository.uonbi.ac.ke/bitstream/handle/11295/95896/N j o g u _ H e a t % 2 0 E q u a t i o n s % 2 0 And%20Their%20Applications%20% 2 8 O n e % 2 0 A n d % 2 0 Tw o % 2 0 Dimension%20Heat%20Equations%29.pdfsequence=1&isAllowed=y
M. Farooq, & A. Salhi. (2013). A preemptive restarting approach to beating inherence instability. Iranian Journal of Science and Technology Transaction a Science, 37(3.1), 349-358. https://doi.org/10.22099/IJSTS.2013.1634.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
