Comparative Analysis of Numerical Methods for Solving 3D Continuation Problem for Wave Equation

Galitdin Bakanov, Sreelatha Chandragiri (), Sergey Kabanikhin and Maxim Shishlenin
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Galitdin Bakanov: Faculty of Natural Sciences, Khoja Akhmet Yassawi International Kazakh-Turkish University, Turkestan 161200, Kazakhstan
Sreelatha Chandragiri: Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
Sergey Kabanikhin: Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia
Maxim Shishlenin: Sobolev Institute of Mathematics, 630090 Novosibirsk, Russia

Mathematics, 2025, vol. 13, issue 18, 1-33

Abstract: In this paper, we develop the explicit finite difference method (FDM) to solve an ill-posed Cauchy problem for the 3D acoustic wave equation in a time domain with the data on a part of the boundary given (continuation problem) in a cube. FDM is one of the numerical methods used to compute the solutions of hyperbolic partial differential equations (PDEs) by discretizing the given domain into a finite number of regions and a consequent reduction in given PDEs into a system of linear algebraic equations (SLAE). We present a theory, and through Matlab Version: 9.14.0.2286388 (R2023a), we find an efficient solution of a dense system of equations by implementing the numerical solution of this approach using several iterative techniques. We extend the formulation of the Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) iterative methods in solving the linear system for computational efficiency and for the properties of the convergence of the proposed method. Numerical experiments are conducted, and we compare the analytical solution and numerical solution for different time phenomena.

Keywords: continuation problem; inverse and ill-posed problem; acoustic wave equation; numerical analysis; regularization; finite difference method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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