SPECTPAL-LIKE RESOLuTION OF FINITE-DIFFERENCE SCHEMES FOR THE HEAT CONDUCTION EQUATION
Abstract
The spectral resolution of the finite-difference theta-method for the heat conduction equation is investigated. By analogy between the frequency response of the heat conduction equation and a low pass filter, we have found an equivalent representation of the finite-difference scheme in the form of two first-order IIR filters with zero group delay. On the basis of the spectral consistency, the error estimate of the discrete filtering model is obtained. The optimal parameters of the IIR filters providing a minimum error of the frequency response within a given spectral range are found. It is remarkable that the optimal ratio of spatial and temporal steps for the theta-method coincides with the ratio provided by the filtering model with co efficients corresponding to a minimum root-mean-square error of the frequency response for the given spectral range. It is shown that the optimized scheme provides a manifold (by a factor of 5–7) reduction in the root-mean-square error of the frequency response in comparison with the 6th order accuracy theta-method. The optimal time step is a little bit larger in comparison with its value in the 6th order accuracy scheme and tends to the last one when the spectral resolution range tends to zero. The obtained results can be used to optimize discretization parameters using the finite-difference methods for the heat conduction equation.
About the Authors
V. M. Volkov
Belarusian State University
Belarus
Belarus
Dr. Sc. (Physics and Mathematics), Assistant Professor, Professor of the Department of Web-Tech nologies and Computer Modeling
A. N. Hureuski
Belarusian State University
Belarus
Belarus
Postgraduate
References
1. Lele S. K. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 1992, vol. 103, no. 1, pp. 16–42. Doi: 10.1016/0021-9991(92)90324-r
2. Tam C. K. W., Webb J. C. Dispersion-relation-preserving finite difference schemes for computational acoustics. Journal of Computational Physics, 1993, vol. 107, no. 2, pp. 262–281. Doi: 10.1006/jcph.1993.1142
3. Volkov V. M. Conservative difference schemes with improved dispersion properties for nonlinear equations of Schrö-dinger type. Differential Equations, 1993, vol. 29, no. 7, pp. 1004–1009.
4. Sergienko A. G. Digital signal processing. Saint Petersburg, BHV-Petersburg Publ., 2011. 768 p. (in Russian).
5. Plura M., Kissing J., Gunkel M., Lenge J., Elbers J.-P., Glingener C., Schulz D., Voges E. Improved split-step method for efficient fiber simulations. Electronics Letters, 2001, vol. 37, no. 5, pp. 286–287. Doi: 10.1049/el:20010212
6. Volkov V. M., Tsiunchik H. S. Split-step method with the use of infinite impulse response filters for solving nonlinear Schrodinger equations. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Be-larus, 2009, vol. 53, no. 5, pp. 22–25 (in Russian).
7. Volkov V. M., Gurevskii A. N., Zhukova I. V. Optimization of compact finite difference schemes with spectral-like resolutionа for the non-stationary Schrodinger equation on the base of digital signal processing methods. Vestnik BGU. Ser. 1. Fizika. Matematika. Informatika = Vestnik BSU. Series 1: Physics. Mathematics. Informatics, 2015, no. 3, pp. 84–89 (in Russian).
8. Samarskii A. A. The theory of Finite Difference schemes. Moscow, Nauka Publ., 1989. 616 p. (in Russian).
9. Saul’ev V. K. About the methods for increasing accuracy and about double-side approximations for solving parabo lic equations. Doklady Akademii nauk SSSR = Doklady of the Academy of Sciences of the USSR, 1958, vol. 118, p. 1088 (in Russian).
Review
Views: 1061
Email this article 