The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method

Gueye, Serigne Bira

doi:10.4236/jemaa.2014.610030

Cited by 8 publications

(14 citation statements)

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“…Depending on the value of the quadruplet of coefficients, nine (9) boundary problems exist: (DD) (ND), (DN), (NN), (RR), (RN), (NR), (DR), and (RD). The first three problems (DD), (ND), and (DN) were solved in [1] and [2]. The problem (NN) leads to a non-regular discretization matrix.…”

Section: General Problemmentioning

confidence: 99%

“…O N [6]. We propose, here, a new method of resolution, faster and more accurate than that of Thomas; as we have already done for the boundary problem of type (DD) [1], and (ND) or (DN) [2]. This method is based essentially on the exact formulation of the inverse of the matrix RR A .…”

Section: ( )mentioning

confidence: 99%

“…Recently, a new method is proposed [1] [2] dealing with this equation in one dimensional case. This new approach, using the finite differences method (FDM), determined the inverse of the matrix obtained from algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

“…This new approach, using the finite differences method (FDM), determined the inverse of the matrix obtained from algebraic equations. However, only the case of boundary conditions Dirichlet-Dirichlet (DD) [1]; Neumann-Dirichlet (ND) and Dirichlet-Neumann (DN) [2] were treated.…”

Section: Introductionmentioning

confidence: 99%

“…First, an inventory is made for possible cases of boundary conditions (DD) (DN), (ND), (RR), (RN), (NR), (DR) and (RD). Then, the cases, which must be solved are identified, considering that the first three cases of boundary problems have been solved in [1] and [2]. Then the cases (RR), (RN), (NR), were treated.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

Gueye¹,

Talla²,

Mbow³

2014

JEMAA

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A new and innovative method for solving the 1D Poisson Equation is presented, using the finite differences method, with Robin Boundary conditions. The exact formula of the inverse of the discretization matrix is determined. This is the first time that this famous matrix is inverted explicitly, without using the right hand side. Thus, the solution is determined in a direct, very accurate (O(h 2 )), and very fast (O(N)) manner. This new approach treats all cases of boundary conditions: Dirichlet, Neumann, and mixed. Therefore, it can serve as a reference for solving the Poisson equation in one dimension.

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Section: General Problemmentioning

confidence: 99%

Section: ( )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

Gueye¹,

Talla²,

Mbow³

2014

JEMAA

Self Cite

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Multi-source off-grid DOA estimation with single snapshot using non-uniform linear arrays

Cao

Wang

et al. 2021

Signal Processing

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A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

Fukuchi

2022

AIP Advances

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The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [( m)].

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The Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method

Cited by 8 publications

References 1 publication

Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

Generalization of the Exact Solution of 1D Poisson Equation with Robin Boundary Conditions, Using the Finite Difference Method

Multi-source off-grid DOA estimation with single snapshot using non-uniform linear arrays

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

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