The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations

The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations Nonlinear wave equation is extensively applied in a wide variety of scientific fields, such as nonlinear optics, solid state physics and quantum field theory. In this paper, two high-performance compact alternating direction implicit (ADI) methods are developed for the nonlinear wave equations. The first scheme is developed a three-level nonlinear difference scheme for nonlinear wave equations, where in x-direction, series of linear tridiagonal systems are solved by Thomas algorithm, while in y-direction, nonlinear algebraic system are computed by Newton’s iterative method. In contrast, the second scheme is linear, and permits the multiple uses of the Thomas algorithm in both x- and y-directions, thus it saves much time cost. By using the discrete energy analysis method, it is shown that both the developed schemes can attain numerical accuracy of order O(τ4+hx4+hy4) in H1-norm. Meanwhile, by the fixed point theorem and symmetric positive-definite properties of coefficient matrix, it is proved that they are both uniquely solvable. Besides, the proposed schemes are extended to the numerical solutions of the coupled sine-Gordon wave equations and damped wave equations. Finally, numerical results confirm the convergence orders and exhibit efficiency of our algorithms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Computation Elsevier

The time fourth-order compact ADI methods for solving two-dimensional nonlinear wave equations

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Publisher
Elsevier
Copyright
Copyright © 2018 Elsevier Inc.
ISSN
0096-3003
eISSN
1873-5649
D.O.I.
10.1016/j.amc.2018.02.010
Publisher site
See Article on Publisher Site

Abstract

Nonlinear wave equation is extensively applied in a wide variety of scientific fields, such as nonlinear optics, solid state physics and quantum field theory. In this paper, two high-performance compact alternating direction implicit (ADI) methods are developed for the nonlinear wave equations. The first scheme is developed a three-level nonlinear difference scheme for nonlinear wave equations, where in x-direction, series of linear tridiagonal systems are solved by Thomas algorithm, while in y-direction, nonlinear algebraic system are computed by Newton’s iterative method. In contrast, the second scheme is linear, and permits the multiple uses of the Thomas algorithm in both x- and y-directions, thus it saves much time cost. By using the discrete energy analysis method, it is shown that both the developed schemes can attain numerical accuracy of order O(τ4+hx4+hy4) in H1-norm. Meanwhile, by the fixed point theorem and symmetric positive-definite properties of coefficient matrix, it is proved that they are both uniquely solvable. Besides, the proposed schemes are extended to the numerical solutions of the coupled sine-Gordon wave equations and damped wave equations. Finally, numerical results confirm the convergence orders and exhibit efficiency of our algorithms.

Journal

Applied Mathematics and ComputationElsevier

Published: Jul 15, 2018

References

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