×

An operator-difference scheme for abstract Cauchy problems. (English) Zbl 1219.65087

Summary: An abstract Cauchy problem for second-order hyperbolic differential equations containing the unbounded self-adjoint positive linear operator \(A(t)\) with domain in an arbitrary Hilbert space is considered. A new second-order difference scheme, generated by integer powers of \(A(t)\), is developed. The stability estimates for the solution of this difference scheme and for the first- and second-order difference derivatives are established in Hilbert norms with respect to space variable. To support the theoretical statements for the solution of this difference scheme, the numerical results for the solution of one-dimensional wave equation with variable coefficients are presented.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
35L90 Abstract hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lamb, H., Hydrodynamics (1974), Cambridge University Press: Cambridge University Press New York · JFM 26.0868.02
[2] Lighthill, J., Waves in Fluids (1978), Cambridge University Press: Cambridge University Press New York · Zbl 0375.76001
[3] Hudson, J. A., The Excitation and Propagation of Elastic Waves (1980), Cambridge University Press: Cambridge University Press New York · Zbl 0435.35002
[4] Jones, D. S., Acoustic and Electromagnetic Waves (1986), Oxford University Press: Oxford University Press New York
[5] Taflove, A., Computational Electrodynamics: The Finite-Difference Time-Domain Method (1995), Artech House: Artech House Boston · Zbl 0840.65126
[6] Mamis, M. S.; Koksal, M., Transient analysis of nonuniform lossy transmission lines with frequency dependent parameters, Electr. Power Syst. Res., 52, 223-228 (1999)
[7] Mamis, M. S.; Koksal, M., Remark on the lumped parameter modeling of transmission lines, Electr. Mach. Power Syst., 28, 566-576 (2000)
[8] Sieniutycz, S.; Berry, R. S., Variational theory for thermodynamics of thermal waves, Phys. Rev. E, 65, 046132-1-046132-11 (2002)
[9] I. Abu-Alshaikh, M.E. Koksal, One-dimensional transient dynamic response in functionally gradient spherical multilayered media, in: Proceedings of Dynamical Systems and Applications, 2004, pp. 1-20.; I. Abu-Alshaikh, M.E. Koksal, One-dimensional transient dynamic response in functionally gradient spherical multilayered media, in: Proceedings of Dynamical Systems and Applications, 2004, pp. 1-20. · Zbl 1341.93042
[10] Abu-Alshaikh, I., One-dimensional wave propagation in functionally graded cylindrical layered media, (Mathematical Methods in Engineering (2007), Springer), 111-121 · Zbl 1130.65096
[11] Liu, G. R.; Han, X.; Lam, K. Y., Stress waves in functionally gradient materials and its use for material characterization, Composites B, 30, 383-394 (1999)
[12] Hillen, T., A turing model with correlated random walk, J. Math. Biol., 35, 49-72 (1996) · Zbl 0863.92001
[13] Heydweiller, J. C.; Sincovec, R. F., Stable difference scheme for solution of hyperbolic equations using method of lines, J. Comput. Phys., 22, 377-388 (1976) · Zbl 0358.65081
[14] Sobolevskii, P. E.; Chebotaryeva, L. M., Approximate solution of the Cauchy problem for an abstract hyperbolic equation by the method of lines, Izv. Vyssh. Uchebn. Zaved. Mat., 180, 103-116 (1977), (in Russian)
[15] Jovanovic, B. S.; Ivanovic, L. D.; Suli, E. E., Convergence of a finite-difference scheme for 2nd-order hyperbolic-equations with variable-coefficients, IMA J. Numer. Anal., 7, 39-45 (1987) · Zbl 0624.65095
[16] Ashyralyev, A.; Sobolevskii, P. E., A note on the difference schemes for hyperbolic equations, Abstr. Appl. Anal., 6, 63-70 (2001) · Zbl 1007.65064
[17] Ashyralyev, A.; Sobolevskii, P. E., (New Difference Schemes for Partial Differential Equations. New Difference Schemes for Partial Differential Equations, Operator Theory: Advances and Applications, vol. 148 (2004), Birkhäuser: Birkhäuser Basel, Boston, Berlin) · Zbl 1060.65055
[18] Ashyralyev, A.; Sobolevskii, P. E., Two new approaches for construction of the high order of accuracy difference schemes for hyperbolic differential equations, Discrete Dyn. Nat. Soc., 2005, 183-213 (2005) · Zbl 1094.65077
[19] Ashyralyev, A.; Koksal, M. E., On the second order of accuracy difference scheme for hyperbolic equations in a Hilbert space, Numer. Funct. Anal. Optim., 26, 739-772 (2005) · Zbl 1098.65055
[20] Ashyralyev, A.; Koksal, M. E., Stability of a second order of accuracy difference scheme for hyperbolic equation in a Hilbert space, Discrete Dyn. Nat. Soc., 2007, 1-25 (2007) · Zbl 1156.65079
[21] Ashyralyev, A.; Koksal, M. E.; Agarwal, Ravi P., A difference scheme for Cauchy problem for hyperbolic equation with self-adjoint operator, Math. Comput. Modelling, 52, 409-424 (2010) · Zbl 1201.65150
[22] Fattorini, H. O., (Second Order Linear Differential Equations in Banach Space. Second Order Linear Differential Equations in Banach Space, North-Holland Mathematics Studies, vol. 107 (1985), North-Holland: North-Holland Amsterdam) · Zbl 0564.34063
[23] Krein, S. G., Linear Differential Equations in a Banach Space (1966), Nauka: Nauka Moscow · Zbl 0636.34056
[24] Gulin, A. V.; Samarskii, A. A., On some results and problems in the stability theory for difference schemes, Mat. Sb., 99, 299-330 (1976)
[25] Samarskii, A. A., Theory of Difference Schemes (1989), Nauka: Nauka Moscow, (in Russian) · Zbl 0971.65076
[26] Samarsky, A. A., Theory of stability and regularization of difference schemes and its application to ill-posed problems of mathematical physics, Appl. Numer. Math., 16, 51-64 (1994) · Zbl 0814.65092
[27] Samarskii, A. A.; Vabischevich, P. N.; Matus, P. P., Difference Schemes with Operator Factors (1998), Institute of Mathematics: Institute of Mathematics Minsk, (in Russian) · Zbl 1189.34107
[28] Jovanovic, B. S.; Vulkov, L. G., On the convergence of difference schemes for hyperbolic problems with concentrated data, SIAM J. Numer. Anal., 41, 516-538 (2003) · Zbl 1050.65082
[29] Piskarev, S., Stability of difference schemes in Cauchy problems with almost periodic solutions, Differ. Uravn., 689-695 (1984), (in Russian)
[30] Mohanty, R. K., An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions, Appl. Math. Comput., 152, 799-806 (2004) · Zbl 1077.65093
[31] Mohanty, R. K., An operator splitting technique for an unconditionally stable difference method for a linear three space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput., 162, 549-557 (2005) · Zbl 1063.65084
[32] Mohanty, R. K., An unconditionally stable finite difference formula for a linear second order one space dimensional hyperbolic equation with variable coefficients, Appl. Math. Comput., 165, 229-236 (2005) · Zbl 1070.65076
[33] Liu, H. W.; Liu, L. B., An unconditionally stable spline difference scheme of \(O(k^2 + h^4)\) for solving the second-order 1D linear hyperbolic equation, Math. Comput. Modelling, 49, 1985-1993 (2009) · Zbl 1171.65424
[34] Ashyralyev, A.; Aggez, N., A note on the difference schemes of the nonlocal boundary value problems for hyperbolic equations, Numer. Funct. Anal. Optim., 25, 439-462 (2004) · Zbl 1065.35021
[35] Ashyralyev, A.; Yildirim, O., On multipoint nonlocal boundary value problems for hyperbolic differential and difference equations, Taiwanese J. Math., 14, 165-194 (2010) · Zbl 1201.65128
[36] Ashyralyev, A.; Yurtsever, H. A., The stability of difference schemes of second-order of accuracy for hyperbolic-parabolic equations, Comput. Math. Appl., 52, 259-268 (2006) · Zbl 1137.65054
[37] Ashyralyev, A.; Ozdemir, Y., Stability of difference schemes for hyperbolic-parabolic equations, Comput. Math. Appl., 50, 1443-1476 (2005) · Zbl 1088.65082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.