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fictitious time integration method of Fundamental Solutions with Chebyshev Polynomials for Solving Poisson-type Nonlinear PDEs

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CMES-COMPUTER MODELING IN ENGINEERING & SCIENCES 2010年第2期56卷 131-151页

作者： Tsai, Chia-Cheng Liu, Chein-Shan Yeih, Wei-Chung Natl Kaohsiung Marine Univ Dept Marine Environm Engn Kaohsiung 811 Taiwan Natl Taiwan Univ Dept Civil Engn Taipei 10764 Taiwan Natl Taiwan Ocean Univ Dept Harbor & River Engn Chilung 202 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Chilung 202 Taiwan

The fictitious time integration method (FTIM) previously developed by Liu and Atluri (2008a) is combined with the method of fundamental solutions and the Chebyshev polynomials to solve Poisson-type nonlinear PDEs. The method of fundamental solutions with Chebyshev polynomials (MFS-CP) is an exponentially-convergent meshless numerical method which is able to solving nonhomogeneous partial differential equations if the fundamental solution and the analytical particular solutions of the considered operator are known. In this study, the MES-CP is extended to solve Poisson-type nonlinear PDEs by using the FTIM. In the solution procedure, the FTIM is introduced to convert a Poisson-type nonlinear PDE into a sequence of linear nonhomogeneous modified Helmholtz equations which are then formally solved by the MFS-CP. Several numerical experiments were carried out to validate the proposed methods.

关键词： fictitious time integration method method of fundamental solutions Chebyshev polynomial Nonlinear partial differential equation

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fictitious time integration method for seeking periodic orbits of nonlinear dynamical systems

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INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS 2023年 152卷

作者： Liu, Chein-Shan Atluri, Satya N. Dai, Honghua Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung 202301 Taiwan Texas Tech Univ Ctr Adv Res Engn Sci Lubbock TX USA Northwestern Polytech Univ Coll Astronaut Xian 710072 Peoples R China

In the paper, we compute the period and periodic orbit of an n-dimensional nonlinear dynamical system by developing two iterative algorithms based on the fictitious time integration method (FTIM). Periodicity condition, also known as the Poincare map, is a necessary condition for the existence of a periodic motion in the state space, which consists of n implicit nonlinear algebraic equations (NAEs). Instead of solving the NAEs by the Newton-Raphson method, we derive an equivalent nonlinear scalar equation to determine unknown period by using the FTIM. The resulting sequence of the iterated periods monotonically converges to a desired period, wherein the unknown initial point on the periodic orbit is determined simultaneously. We find that the second iterative algorithm using the matrix shape function to convert the periodic problem into a corresponding initial value problem is convergent faster than the first iterative algorithm. A key point is that the periodicity condition is satisfied automatically by the second iterative algorithm when the terminal values of the new variables are convergent. Numerical examples exhibit some major advantages of these two iterative algorithms.

关键词： Nonlinear dynamical system Periodic orbit Matrix shape function fictitious time integration method Iterative algorithm

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CONSTRAINT-TYPE fictitious time integration method FOR SOLVING NON-LINEAR MULTI-DIMENSIONAL ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

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JOURNAL OF MARINE SCIENCE AND TECHNOLOGY-TAIWAN 2020年第3期28卷 168-178页

作者： Chen, Yung-Wei Liu, Chein-Shan Chang, Yen-Shen Chang, Jiang-Ren Natl Taiwan Ocean Univ Dept Marine Engn Keelung Taiwan Natl Taiwan Ocean Univ Ctr Excellence Ocean Engn Keelung Taiwan Natl Taiwan Ocean Univ Dept Syst Engn & Naval Architecture Keelung Taiwan

In this paper, we propose a constraint-type fictitious time integration method (FTIM) for solving multi-dimensional non-linear elliptic-type partial differential equations. Based on the variable transformation of FTIM, the original governing equation is transformed into a new parabolic equation of an evolution type by introducing a space-time variable, and a new time integration direction is obtained. However, the space-time variable depends on the governing equation, boundary condition and fictitious time variable, especially due to the nonlinear effect. Previous studies have not discussed the definition of these nonlinear parameter problems, which may result in severe numerical instability and inaccuracy. To completely overcome this nonlinear parameter problem, a space-time variable with a minimum fictitious time size is introduced into the algorithm. By imposing a constraint condition that involves the system energy in the space domain and the minimum fictitious time step, the proposed scheme can absolutely satisfy the stringent convergence criterion and can quickly approach the true solution, even under a very small time step. More importantly, the convergence speed depends only on a space-time variable. The accuracy and efficiency of the scheme are evaluated by comparing the estimation results with those of previous studies. The obtained results demonstrate that the proposed method efficiently finds the true solution and can significantly improve both the accuracy and convergence.

关键词： algebraic equations ordinary differential equation fictitious time integration method Newton's method iterative scheme

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USING fictitious time integration method TO STUDY WAVE PROPAGATION OVER ARBITRARY BATHYMETRY

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JOURNAL OF MECHANICS 2013年第3期29卷 551-558页

作者： Chang, J-Y Tsai, C-C Hsu, T-W Tainan Univ Technol Dept Business Adm Tainan 71002 Taiwan Natl Kaohsiung Marine Univ Dept Marine Environm Engn Kaohsiung 81157 Taiwan Natl Cheng Kung Univ Dept Hydraul & Ocean Engn Tainan 70101 Taiwan

In this study, the fictitious time integration method (FTIM) is applied to investigate wave propagation over an arbitrary bathymetry with measured uncertainty. The FTIM is used to convert the higher-order elliptic mild-slope equation (EMSE) into a FTIM like EMSE (FtimeMSE). It has the advantage to describe wave transformation from deep water to shallow water region in a large coastal area with numerical efficiency. The validity of the noise resistance for the measured uncertainty of the bathymetry is also studied. In addition, typical examples for waves propagating over an elliptic shoal rest on a horizontal and sloping bottom is presented. It is concluded that the FTIM is robust in the numerical stability and capable of against the noise of the measurement.

关键词： fictitious time integration method Mild slope equation Water wave

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Highly Accurate Golden Section Search Algorithms and fictitious time integration method for Solving Nonlinear Eigenvalue Problems

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Computer Modeling in Engineering & Sciences 2024年第5期139卷 1317-1335页

作者： Chein-Shan Liu Jian-Hung Shen Chung-Lun Kuo Yung-Wei Chen Center of Excellence for Ocean Engineering Taiwan Ocean UniversityKeelung202301TaiwanChina Department of Marine Engineering Taiwan Ocean UniversityKeelung202301TaiwanChina

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue *** each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex ***,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation *** real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very *** the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

关键词： Nonlinear eigenvalue problem quadratic eigenvalue problem two new merit functions golden section search algorithm fictitious time integration method

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Applications of the local RBF collocation method and the fictitious time integration method for Burgers' equations 37

Applications of the local RBF collocation method and the fic...

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37th National Conference on Theoretical and Applied Mechanics (37th-NCTAM) / 1st International Conference on Mechanics (1st-ICM)

作者： Fan, Chia-Ming Yang, Chi-Hung Gu, Meng-Huang Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan Natl Ctr Res Earthquake Engn Natl Appl Res Labs Taipei Taiwan

In this paper, a meshfree numerical scheme, which is a combination of the local RBF (radial basis function) collocation method (LRBFCM) and the fictitious time integration method (FTIM), is proposed to efficiently and accurately analyze the twodimensional Burgers' equations. The Burgers' equations, which can be regarded as the simplified form of the Navier-Stokes equations, are a system of time-and space-dependent nonlinear partial differential equations, so it is very difficult and important to be accurately solved. The LRBFCM, one of newly-developed meshfree methods, is adopted for spatial discretization and, at the meantime, the implicit Euler method is used for temporal discretization. Then, the system of nonlinear algebraic equations at every time step will be efficiently resolved by the FTIM, which can avoid the calculation of the Jacobian matrix. The numerical results and comparisons of one numerical example will be provided to validate the accuracy and the efficiency of the proposed meshfree numerical scheme. (C) 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license.

关键词： meshfree numerical scheme local RBF collocation method fictitious time integration method Burgers' equations.

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Applications of the Local RBF Collocation method and the fictitious time integration method for Burgers’ Equations

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Procedia Engineering 2014年 79卷 569-574页

作者： Chia-Ming Fan Chi-Hung Yang Meng-Huang Gu Department of Harbor and River Engineering & Computation and Simulation Center National Taiwan Ocean University Keelung 20224 Taiwan National Center for Research on Earthquake Engineering National Applied Research Laboratories Taipei Taiwan

In this paper, a meshfree numerical scheme, which is a combination of the local RBF (radial basis function) collocation method (LRBFCM) and the fictitious time integration method (FTIM), is proposed to efficiently and accurately analyze the two-dimensional Burgers’ equations. The Burgers’ equations, which can be regarded as the simplified form of the Navier-Stokes equations, are a system of time-and space-dependent nonlinear partial differential equations, so it is very difficult and important to be accurately solved. The LRBFCM, one of newly-developed meshfree methods, is adopted for spatial discretization and, at the meantime, the implicit Euler method is used for temporal discretization. Then, the system of nonlinear algebraic equations at every time step will be efficiently resolved by the FTIM, which can avoid the calculation of the Jacobian matrix. The numerical results and comparisons of one numerical example will be provided to validate the accuracy and the efficiency of the proposed meshfree numerical scheme.

关键词： meshfree numerical scheme local RBF collocation method fictitious time integration method Burgers’ equations.

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Generalized finite difference method for solving two-dimensional non-linear obstacle problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2013年第9期37卷 1189-1196页

作者： Chan, Hsin-Fang Fan, Chia-Ming Kuo, Chia-Wen Natl Taiwan Ocean Univ Dept Harbor & River Engn Keelung 20224 Taiwan Natl Taiwan Ocean Univ Computat & Simulat Ctr Keelung 20224 Taiwan

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer-Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Obstacle problems Generalized finite difference method fictitious time integration method Meshless method Non-linear free boundary problems

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Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line

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JOURNAL OF COMPUTATIONAL PHYSICS 2016年 316卷 10-20页

作者： Hashemi, M. S. Baleanu, D. Univ Bonab Basic Sci Fac Dept Math POB 55517-61167 Bonab Iran Cankaya Univ Dept Math Ogretmenler Caddesi 14 TR-06530 Ankara Turkey Inst Space Sci Magurele Romania

We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate v is imposed onto the problem in order to transform the dependent variable u(x, t) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation. (C) 2016 Elsevier Inc. All rights reserved.

关键词： time-fractional telegraph equation Caputo derivative fictitious time integration method Group preserving scheme method of line

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Study on the detection of groundwater boundary based on the Trefftz method

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NATURAL HAZARDS 2024年第8期120卷 8057-8085页

作者： Su, Yan Huang, Bin Yang, Lingjun Lai, Xiaohe Lin, Chuan Xie, Xiudong Hong, Lidan Fuzhou Univ Coll Civil Engn Fuzhou 350116 Fujian Peoples R China Fujian Univ Technol Inst Smart Marine & Engn Fuzhou 350118 Fujian Peoples R China

Detecting water head is crucial in groundwater utilization, and requires quick and accurate solutions. This study employs a method combining the collocation Trefftz method (CTM) and the fictitious time integration method (FTIM) for groundwater head detection and restoration. The Laplace equation is solved using CTM, and the Trefftz basis function is linearly combined to fit the exact solution while adding characteristics length for numerical stability. The FTIM solves the nonlinear algebraic equations system for water head detection. Numerical examples quantify the method's accuracy, and boundary restoration results are compared with the Picard successive approximation method, showcasing FTIM's advantages in convergence steps and precision. The solution comparison under irregular boundary conditions further verifies the proposed method's efficacy (MAE <= 10-15). The CTM-FTIM calculated water level boundary aligns with the actual boundary, and its noise immunity is verified using real observation well data (MAE <= 10-2, Itertimes <= 5000). The CTM-FTIM method eliminates meshing needs for quick solutions in irregular regions, accurately determining water levels in the study domain using few known boundary points, solving infinite domain groundwater head detection.

关键词： fictitious time integration method Groundwater head detection recovery Noise immunity Observation well data Trefftz method

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