Error estimate of a Legendre-Galerkin Chebyshev collocation method for a class of parabolic inverse problem

The accurate calculation of the sound field is one of the most concerning issues in hydroacoustics. The one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it is difficult to solve actual ocean acoustic fields using this model due to its application conditions and approximation error. Therefore, it is necessary to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation without using simplified models. Here, two commonly used spectral methods, Chebyshev–Galerkin and Chebyshev–collocation, are used to correctly solve the two-dimensional Helmholtz model equation. Since Chebyshev–collocation does not require harsh boundary conditions for the equation, it is then used to solve ocean acoustic propagation. The numerical calculation results are compared with analytical solutions to verify the correctness of the method. Compared with the mature Kraken program, the Chebyshev–collocation method exhibits higher numerical calculation accuracy. Therefore, the Chebyshev–collocation method can be used to directly solve the representative two-dimensional ocean acoustic propagation equation. Because there are no model constraints, the Chebyshev–collocation method has a wide range of applications and provides results with high accuracy, which is of great significance in the calculation of realistic ocean sound fields.

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Capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by Chebyshev collocation method

Journal of Computational Physics ◽

10.1016/j.jcp.2012.12.012 ◽

2013 ◽

Vol 238 ◽

pp. 210-216 ◽

Cited By ~ 6

Author(s):

Jonathan Hagan ◽

Jānis Priede

Keyword(s):

Collocation Method ◽

Hydrodynamic Stability ◽

Chebyshev Collocation Method ◽

Stability Problems ◽

Chebyshev Collocation ◽

Capacitance Matrix

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The impact of the Chebyshev collocation method on solutions of the time-fractional Black–Scholes

Mathematical Sciences ◽

10.1007/s40096-020-00357-2 ◽

2020 ◽

Author(s):

H. Mesgarani ◽

A. Beiranvand ◽

Y. Esmaeelzade Aghdam

Keyword(s):

Collocation Method ◽

Linear Interpolation ◽

Stability And Convergence ◽

Finite Domain ◽

European Options ◽

Chebyshev Collocation Method ◽

Chebyshev Collocation ◽

Black Scholes ◽

Temporal Derivative ◽

The Impact

AbstractThis paper presents a numerical solution of the temporal-fractional Black–Scholes equation governing European options (TFBSE-EO) in the finite domain so that the temporal derivative is the Caputo fractional derivative. For this goal, we firstly use linear interpolation with the $$(2-\alpha)$$ ( 2 - α ) -order in time. Then, the Chebyshev collocation method based on the second kind is used for approximating the spatial derivative terms. Applying the energy method, we prove unconditional stability and convergence order. The precision and efficiency of the presented scheme are illustrated in two examples.

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