4.7 Article

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

Journal

APPLIED MATHEMATICS AND COMPUTATION
Volume 190, Issue 2, Pages 1683-1690

Publisher

ELSEVIER SCIENCE INC
DOI: 10.1016/j.amc.2007.02.097

Keywords

hyperbolic equations; first-order space derivative; explicit finite difference scheme; stability interval; pade' approximation; first time step

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in this piece of work, 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 u(n) = a(x, t)u(xx) + alpha(x, t)u(x) - 2 eta(2) (x, t) u, u(n) = 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, and u(n) = 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, 0 < x,y,z < 1,t > 0 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. (C) 2007 Elsevier Inc. All rights reserved.

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