×

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms. (English) Zbl 1122.65381

Summary: We introduce a new idea and obtain stability interval for explicit difference schemes of \(O(k^2+h^2)\) for one, two and three space dimensional second-order hyperbolic equations
\[ \begin{aligned} u_{tt}&= a(x,t)u_{xx}+ \alpha(x,t)u_x- 2\eta^2(x,t)u,\\ u_{tt}&=a(x,y,t)u_{xx}+ b(x,y,t)u_{yy}+ \alpha(x,y,t)u_x+ \beta(x,y,t)u_y- 2\eta^2(x,y,t)u,\end{aligned} \] and
\[ \begin{aligned} u_{tt}=&a(x,y,z,t)u_{xx}+ b(x,y,z,t)u_{yy}+ c(x,y,z,t)u_{zz}+\alpha(x,y,z,t)u_x+\\ +&\beta(x,y,z,t)u_y+\gamma(x,y,z,t)u_z- 2\eta^2(x,y,z,t)u, \quad 0<x,y,z<1,\;t>0, \end{aligned} \]
subject to appropriate initial and Dirichlet boundary conditions, where \(h>0\) and \(k>0\) are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of \(O(k^2)\) in order to obtain numerical solution of \(u\) at first time step in a different manner.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mohanty, R. K.; Arora, U., A new discretization method of order four for the numerical solution of one space dimensional second order quasi-linear hyperbolic equation, Int. J. Math. Edu. Sci. Technol., 33, 829-838 (2002)
[2] Mohanty, R. K.; Arora, U.; Jain, M. K., Linear stability analysis and fourth order approximations at first time level for the two space dimensional mildly quasi-linear hyperbolic equations, Numer. Methods. Partial Differen. Equat., 17, 607-618 (2001) · Zbl 0990.65102
[3] Mohanty, R. K.; Arora, U.; Jain, M. K., Fourth order approximation for the three space dimensional certain mildly quasi-linear hyperbolic equation, Numer. Methods Partial Differen. Equat., 17, 277-289 (2001) · Zbl 0982.65096
[4] Ciment, M.; Leventhal, S. H., Higher order compact implicit schemes for the wave equation, Math. Comput., 29, 985-994 (1975) · Zbl 0309.35043
[5] Ciment, M.; Leventhal, S. H., A note on the operator compact implicit method for the wave equation, Math. Comput., 32, 143-147 (1978) · Zbl 0373.35039
[6] Twizell, E. H., An explicit difference method for the wave equation with extended stability range, BIT, 19, 378-383 (1979) · Zbl 0441.65066
[7] Greenspan, D., Approximate solution of initial boundary wave equation problems by boundary value technique, Commun. ACM, 11, 760-763 (1968) · Zbl 0176.15304
[8] Strickwerda, J., Finite Difference Schemes and Partial Differential Equations (1989), Wadsworth and Books: Wadsworth and Books London
[9] Roisin, B. C., Analytical linear stability criteria for the Leap-Frog, Dufort-Frankel method, J. Comput. Phys., 53, 227-239 (1984) · Zbl 0571.76091
[10] Mohanty, R. K.; Jain, M. K.; George, K., Fourth order approximation at first time level, linear stability analysis and the numerical solution of multi-dimensional second order non-linear hyperbolic equations in polar coordinates, J. Comput. Appl. Math., 93, 1-12 (1998) · Zbl 0932.65092
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.