Abstract
A method is developed which superimposes a uniform grid of step-sizeh on the space variablex in the wave equation∂ 2u/∂x2=∂2u/∂t2. The resulting system of second order ordinary differential equations is solved using a rational approximant toe lA, wherel is the time step andA is the coefficient matrix. A seven point explicit finite difference scheme is derived whose consistency, stability and convergence are discussed. The rational approximant is seen to have a stability range of 0 ≦l/h=r≦√3. Numerical results of the algorithm applied to two problems, one of which has a discontinuity between the initial and boundary conditions, are reported and compared with the familiar five point explicit scheme, which may be derived using the same approach with a different rational approximant and whose stability range is 0≦r≦1.
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References
G. Fairweather,A note on the efficient implementation of certain Padé methods for linear parabolic problems, BIT 18:1 (1978), 106–109.
J. D. Lawson and J. Ll. Morris,The extrapolation of first order parabolic partial differential equations I, SIAM J. Num. Anal. 15:6 (1978), 1212–1224.
J. D. Lawson and J. Ll. Morris,A note on the efficient implementation of splitting methods in two space variables, BIT 17:4 (1977), 492–493.
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Twizell, E.H. An explicit difference method for the wave equation with extended stability range. BIT 19, 378–383 (1979). https://doi.org/10.1007/BF01930991
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DOI: https://doi.org/10.1007/BF01930991