Application of Direct Newton’s Methods for Efficient Solution to Convective-Diffusive Transport Processes

2005 ◽  
Author(s):  
Mandhapati P. Raju
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Simple Procedure ◽  
Linear Equations ◽  
Computation Time ◽  
Transport Processes ◽  
Diffusive Transport ◽  
Direct Solution ◽  
Solution Methods ◽  
Direct Solution Method

Newton’s iterative technique is commonly used in solving a system of non-linear equations. The advantage of using Newton’s method is that it gives local quadratic convergence leading to high computational efficiency. Specifically, Newton’s method has been applied to finite volume formulation for convective-diffusive transport processes. A direct solution method is adopted. Development of sparse direct solvers has significantly reduced the computation time of direct solution methods. Here UMFPACK (Unsymmetric Multi-Frontal method), has been used to solve the resulting linear system obtained from Newton’s step. A simple damping strategy is applied to ensure the global convergence of the system of equations during the first few iterations. The efficiency of this method is compared to that of Picard’s iterative procedure and the SIMPLE procedure for convective-diffusive transport processes. A modified Newton technique is also analyzed which lead to significant reduction in total CPU time.

Quantum Newton’s Method for Solving the System of Nonlinear Equations

SPIN ◽  
2021 ◽  
pp. 2140004
Author(s):  
Cheng Xue ◽  
Yuchun Wu ◽  
Guoping Guo
Keyword(s):  
Quantum Computing ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Linear Equations ◽  
Data Conversion ◽  
System Of Nonlinear Equations ◽  
Dimensional System ◽  
System Of Linear Equations ◽  
Whole Process

While quantum computing provides an exponential advantage in solving the system of linear equations, there is little work to solve the system of nonlinear equations with quantum computing. We propose quantum Newton’s method (QNM) for solving [Formula: see text]-dimensional system of nonlinear equations based on Newton’s method. In QNM, we solve the system of linear equations in each iteration of Newton’s method with quantum linear system solver. We use a specific quantum data structure and [Formula: see text] tomography with sample error [Formula: see text] to implement the classical-quantum data conversion process between the two iterations of QNM, thereby constructing the whole process of QNM. The complexity of QNM in each iteration is [Formula: see text]. Through numerical simulation, we find that when [Formula: see text], QNM is still effective, so the complexity of QNM is sublinear with [Formula: see text], which provides quantum advantage compared with the optimal classical algorithm.

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Nenzi Wang ◽  
Shih-Hung Chang ◽  
Hua-Chih Huang
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Compressible Fluid ◽  
Newton's Method ◽  
Reynolds Equation ◽  
Multigrid Method ◽  
Iterative Solution ◽  
Sor Method ◽  
Solution Methods ◽  
Iterative Solution Methods

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

Cascadic Newton’s method for the elliptic Monge–Ampère equation

2020 ◽  
Vol 18 (03) ◽  
pp. 2050018
Author(s):  
Qin Li ◽  
Zhiyong Liu
Keyword(s):  
Finite Difference ◽  
Newton’S Method ◽  
Newton's Method ◽  
Finite Difference Methods ◽  
Computation Time ◽  
Type Interpolation ◽  
Local Type ◽  
Cascadic Multigrid ◽  
Monge Ampere ◽  
Prolongation Operator

In this paper, a cascadic Newton’s method is designed to solve the Monge–Ampère equation. In the process of implementing the cascadic multigrid, we use the Full-Local type interpolation as prolongation operator and Newton iteration as smoother. In order to obtain Full-Local type interpolation, we provide several finite difference stencils. Especially, the skewed finite difference methods are first applied by us for the elliptic Monge–Ampère equation. Based on Full-Local interpolation techniques and cascade principle, the new algorithm can save a large amount of computation time. Some numerical experiments are provided to confirm the efficiency of our proposed method.

Efficiency of Solution Methods for Kepler’s Equation

2016 ◽  
Vol 851 ◽  
pp. 587-592
Author(s):  
João Francisco Nunes de Oliveira ◽  
Roberta Veloso Garcia ◽  
Hélio Koiti Kuga ◽  
Estaner Claro Romão
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Kepler's Equation ◽  
Eccentric Orbits ◽  
Kepler’S Equation ◽  
Solution Methods ◽  
Number Of Iterations

This article discusses, in the case of eccentric orbits, some solution methods for Kepler's equation, for instance: Newton's method, Halley method and the solution by Fourire-Bessel expansion. The efficiency of solution methods is evaluated according to the number of iterations that each method needs to lead to a solution within the specified tolerance. The solution using Fourier-Bessel series is not an iterative method, however, it was analyzed the number of terms required to achieve the accuracy of the prescribed solution.

scholarly journals DEVELOPMENT OF NEW ITERATIVE SCHEME FOR SOLVING NON-LINEAR EQUATIONS

2021 ◽  
Author(s):  
Tusar singh ◽  
Dwiti Behera
Keyword(s):  
Iterative Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Iterative Scheme ◽  
Linear Equations ◽  
Quadrature Rule ◽  
Modified Newton’S Method ◽  
Newton Raphson ◽  
Raphson Method ◽  
Non Linear Equations

Within our study a special type of 〖iterative method〗^ω is developed by upgrading Newton-Raphson method. We have modified Newton’s method by using our newly developed quadrature rule which is obtained by blending Trapezoidal rule and open type Newton-cotes two point rule. Our newly developed method gives better result than the Newton’s method. Order of convergence of our newly discovered quadrature rule and iterative method is 3.

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