A Chebyshev Spectral Method for Normal Mode and Parabolic Equation Models in Underwater Acoustics

In this paper, the Chebyshev spectral method is used to solve the normal mode and parabolic equation models of underwater acoustic propagation, and the results of the Chebyshev spectral method and the traditional finite difference method are compared for an ideal fluid waveguide with a constant sound velocity and an ideal fluid waveguide with a deep-sea Munk speed profile. The research shows that, compared with the finite difference method, the Chebyshev spectral method has the advantages of a high computational accuracy and short computational time in underwater acoustic propagation.

Download Full-text

Multidomain Chebyshev spectral method for 3-D dc resistivity modeling

Geophysics ◽

10.1190/1.1444080 ◽

1996 ◽

Vol 61 (6) ◽

pp. 1616-1623 ◽

Cited By ~ 6

Author(s):

Shengkai Zhao ◽

Matthew J. Yedlin

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Spectral Method ◽

Direct Current ◽

Chebyshev Polynomials ◽

Difference Method ◽

Dc Resistivity ◽

Chebyshev Spectral Method ◽

The Finite Difference Method ◽

Resistivity Modeling

We use the multidomain Chebyshev spectral method to solve the 3-D forward direct current (dc) resistivity problem. We divided the whole domain into a number of subdomains and approximate the potential function by a separate set of Chebyshev polynomials in each subdomain. At an interface point, we require that both the potential and the flux be continuous. Numerical results show that for the same accuracy the multidomain Chebyshev spectral method is 2 to 260 times faster than the finite‐difference method.

Download Full-text

A Two-Grid Block-Centered Finite Difference Method for the Nonlinear Time-Fractional Parabolic Equation

Journal of Scientific Computing ◽

10.1007/s10915-017-0380-4 ◽

2017 ◽

Vol 72 (2) ◽

pp. 863-891 ◽

Cited By ~ 14

Author(s):

Xiaoli Li ◽

Hongxing Rui

Keyword(s):

Parabolic Equation ◽

Finite Difference ◽

Finite Difference Method ◽

Difference Method ◽

Grid Block

Download Full-text

A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

Journal of Algorithms & Computational Technology ◽

10.1177/17483026211008409 ◽

2021 ◽

Vol 15 ◽

pp. 174830262110084

Author(s):

Xianjuan Li ◽

Yanhui Su

Keyword(s):

Finite Difference ◽

Finite Difference Method ◽

Differential Equations ◽

Spectral Method ◽

Fractional Differential Equations ◽

Difference Method ◽

Spectral Collocation ◽

Parallel Method ◽

Fractional Differential ◽

Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

Download Full-text