A Complete Procedure for a Constraint-Type Fictitious Time Integration Method to Solve Nonlinear Multi-Dimensional Elliptic Partial Differential Equations

Yung-Wei Chen, Jian-Hung Shen (), Yen-Shen Chang and Ching-Chuan Tan
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Yung-Wei Chen: Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Jian-Hung Shen: Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Yen-Shen Chang: Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan
Ching-Chuan Tan: Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan

Mathematics, 2023, vol. 11, issue 1, 1-15

Abstract: In this paper, an efficient and straightforward numerical procedure is constructed to solve multi-dimensional linear and nonlinear elliptic partial differential equations (PDEs). Although the numerical procedure for the constraint-type fictitious time integration method overcomes the numerical stability problem, the parameter’s definition, numerical accuracy and computational efficiency have not been resolved, and the lack of initial guess values results in reduced computational efficiency. Therefore, the normalized two-point boundary value solution of the Lie-group shooting method is proposed and considered in the numerical procedure to avoid the problem of the initial guess value. Then, a space-time variable, including the minimal fictitious time step and convergence rate factor, is introduced to study the relationship between the initial guess value and convergence rate factor. Some benchmark numerical examples are tested. As the results show, this numerical procedure using the normalized boundary value solution can significantly converge within one step, and the numerical accuracy is better than that demonstrated in the previous literature.

Keywords: boundary solution integration; nonlinear partial differential equations; fictitious time integration method; constraint-type fictitious time integration method; traditional gradient convergence method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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