Abstract
We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.
Highlights
We consider the one-dimensional wave equation in polar forms: utt = urr + D (r) ur + E (r) u + f (r, t), 0 < r < 1, t > 0, (1)with the following initial conditions:u (r, 0) = φ (r), ut (r, 0) = φ (r), 0 ≤ r ≤ 1, (2)and the following boundary conditions:u (0, t) = p0 (t), u (1, t) = p1 (t), t ≥ 0, (3)where D(r) = γ/r and E(r) = −γ/r2
We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in r-direction and finite difference approximation in t-direction for the numerical solution of one-dimensional wave equation in polar coordinates
The study of wave equation in polar form is of keen interest in the fields like acoustics, electromagnetic, fluid dynamics, mathematical physics, and so forth
Summary
In this paper, using nine grid points (see Figure 1), we discuss a new three-level implicit non-polynomial spline finite difference method of accuracy two in time and four in space for the solution of one-dimensional wave equation in polar forms.
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