Open Access

ARTICLE

Approach for the Simulation of Linear PDEs with Constant Coefficients, Testing Multi-Dimensional Helmholtz and Wave Equations

Chein-Shan Liu, Chung-Lun Kuo^*
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 202301, Taiwan
* Corresponding Author: Chung-Lun Kuo. Email:
(This article belongs to the Special Issue: Advances in Methods of Computational Modeling in Engineering Sciences, a Special Issue in Memory of Professor Satya Atluri)

Digital Engineering and Digital Twin https://doi.org/10.32604/dedt.2024.042804

Received 13 June 2023; Accepted 05 December 2023; Published online 15 August 2024

Abstract

A new concept of projective solution is introduced for the second-order linear partial differential equations (PDEs) endowed with constant coefficients. In terms of a projective variable the PDE is transformed to a second-order ordinary differential equation (ODE) with constant coefficients at the first time. The characteristic form appears as the coefficient preceding the second-order derivative term. Depending on the characteristic form and coefficients we can derive various parameters-dependent particular solutions, which can be adopted as the bases to expand the solution. The Helmholtz and wave equations are solved by the projection method. We project the field point to a unit characteristic vector to obtain a constant ODE, whose two linearly independent projective solutions are cosine and sine functions. When we expand the solution in terms of these functions as the bases, we can create a powerful numerical technique to solve the Helmholtz equations with high accuracy, even the wave number is quite large. We extend the results to the multi-dimensional wave equation, whose g-analytic function theory and the Cauchy-Riemann equations are deduced. We derive an effective and simple projective solutions method (PSM) used in the computations, which outperforms the conventional methods. Numerical experiments indeed verify the accuracy and efficiency of the PSM.

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Keywords

Characteristic form; characteristic vector; Helmholtz equations; projective solution method; wave equations; g-analytic function theory; g-analytic Cauchy-Riemann equations

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