Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem

In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Gree...

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Vydáno v:Numerical methods for partial differential equations Ročník 40; číslo 3
Hlavní autoři: Li, Hu, Huang, Jin
Médium: Journal Article
Jazyk:angličtina
Vydáno: Hoboken, USA John Wiley & Sons, Inc 01.05.2024
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Témata:
Accuracy
Algorithms
Asymptotic series
axisymmetric Laplace equation
Boundary value problems
Extrapolation
extrapolation algorithm
Formulas (mathematics)
Integral equations
Laplace equation
Linear operators
Numerical analysis
Operators (mathematics)
Quadratures
the quadrature method
ISSN:0749-159X, 1098-2426
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Shrnutí:In order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of O(h3)$$ O\left({h}^3\right) $$ and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power O(h3)$$ O\left({h}^3\right) $$ is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of O(h5)$$ O\left({h}^5\right) $$. Finally, two numerical examples are presented to demonstrate the efficiency of the method.
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ISSN:0749-159X
1098-2426
DOI:10.1002/num.23079