Finite line method for solving high-order partial differential equations in science and engineering

Gao, Xiao‐Wei; Zhu, Yu-Mo; Pan, Tao

doi:10.1016/j.padiff.2022.100477

Cited by 7 publications

(8 citation statements)

References 44 publications

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“…When grid cells are equal in size, truncation errors (errors from approximating derivatives) are evenly distributed, enhancing the overall solution accuracy. On the other hand, non-uniform grids can lead to varying truncation errors that complicate error analysis and reduce accuracy [22]. Furthermore, many numerical methods, including the FDM, have stability requirements dependent on grid spacing, and a uniform grid helps ensure this stability.…”

Section: Application and Results: Charge In Free Spacementioning

confidence: 99%

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Salem,

Aldabbagh

2024

Mathematics

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Poisson’s equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson’s equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution.

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Section: Application and Results: Charge In Free Spacementioning

confidence: 99%

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Salem,

Aldabbagh

2024

Mathematics

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“…The line-based methods include the conventional finite difference method (FDM) [7] and the recently proposed finite line method (FLM) [8]. In these methods, the computational domain is discretized into a series of points and lines to compute the spatial partial derivatives used in the PDEs are formed by around points.…”

Section: Line-based Methodsmentioning

confidence: 99%

“…To solve the PDEs with a set of properly posed BC as listed above, a number of numerical methods are available, which can be globally divided into two big categories according to the geometry discretization and operation dimensions: the full dimensionality methods including the finite element method [3], finite block method [4], element differential method [2], and so on, and mesh reduction methods including surface-based methods (i.e., the boundary element method [5], finite volume method [6], etc. ), the line-based methods (i.e., the finite difference method [7], finite line method [8], etc. ), and the point-based methods (i.e., the mesh free method [9], free element method [10], fundamental solution method [11], etc.).…”

Section: Introductionmentioning

confidence: 99%

Progress of Mesh Reduction Methods

GAO,

LIU,

FAN

2023

Boundary Elements and Other Mesh Reduction Methods XLVI

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According to the discretization manner and manipulation dimensionality, numerical methods can be classified into two big groups, the full dimensionality method and mesh reduction methods. The former includes the finite element method, finite block method, element differential method and so on, and the latter can be further divided into three types: surface-based, line-based and point-based methods. The surface-based method consists of the boundary element method, finite volume method, boundary face method and so on; the line-based method includes the finite difference method, finite element line method, finite line method and so on; and the point-based method mainly consists of the mesh free method, fundamental solution method, free element method and so on. The paper gives a detailed description on the progress achieved in recent years in the mesh reduction methods, among which a brief introduction will go into the surface-and point-based methods and a detailed one will be given to the line-based methods, especially the finite line method (FLM). FLM is a new type collocation method, which has a distinct feature that the solution scheme to solve a partial differential equation is established by using two or three lines crossing a collocation point for 2D or 3D problems. The Lagrange polynomial formulation is used to approximate the variation of the coordinates and physical variables over each line. And the directional derivative technique is used to derive various orders spatial partial derivatives of any physical variables. The derived partial derivatives can be directly substituted into the governing partial differential equations and related boundary conditions to set up the discretized system of equations. An example will be given for the mechanical problem to demonstrate the accuracy and efficiency of the descripted mesh reduction methods.

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“…is the truncation error caused by one step of iteration which is called local truncation error. Hence, the j th local absolute maximum error 3 . Te global truncation error TE j is the cumulative truncation error up to j th iteration.…”

Section: Semidiscretizationmentioning

confidence: 99%

“…where ε and μ are "small" positive interrelated parameters simultaneously approaching zero and L 1 , L 2 , and L 3 are linear diferential operators whose orders are l 1 > l 2 > l 3 , respectively, for a sufciently smooth function φ. Tese kinds of equations arise in the felds of fuid dynamics, quantum mechanics, chemical fow reactor theory, and DC motor analysis [2,3]. Te two-parameter singularly perturbed parabolic differential equation is the focus of some recent studies [4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

Worku,

Duressa

2023

International Journal of Mathematics and Mathematical Sciences

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A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

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Finite line method for solving high-order partial differential equations in science and engineering

Cited by 7 publications

References 44 publications

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Progress of Mesh Reduction Methods

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

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