A higher order frozen Jacobian iterative method for solving Hamilton-Jacobi equations (original) (raw)
Related papers
Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved. Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.
Fixed-point Iterative Sweeping Methods for Static Hamilton-Jacobi Equations
Methods and Applications of Analysis, 2006
Fast sweeping methods utilize the Gauss-Seidel iterations and alternating sweeping strategy to achieve the fast convergence for computations of static Hamilton-Jacobi equations. They take advantage of the properties of hyperbolic PDEs and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. The time-marching approach to steady state calculation is much slower than the fast sweeping methods due to the CFL condition constraint. But this kind of fixed-point iterations as time-marching methods have explicit form and do not involve inverse operation of nonlinear Hamiltonian. So it can solve general Hamilton-Jacobi equations using any monotone numerical Hamiltonian and high order approximations easily. In this paper, we adopt the Gauss-Seidel idea and alternating sweeping strategy to the time-marching type fixed-point iterations to solve the static Hamilton-Jacobi equations. Extensive numerical examples verify at least a 2 ∼ 5 times acceleration of convergence even on relatively coarse grids. The acceleration is even more when the grid is further refined. Moreover the Gauss-Seidel philosophy and alternating sweeping strategy improves the stability, i.e., a larger CFL number can be used. Also the computational cost is exactly the same as the time-marching scheme at each time step.
A PRECONDITIONED JACOBI-TYPE METHOD FOR SOLVING MULTI-LINEAR SYSTEMS
Journal of Mahani Mathematical Research, 2021
Recently, Zhang et al. [Applied Mathematics Letters 104 (2020) 106287] proposed a preconditioner to improve the convergence speed of three types of Jacobi iterative methods for solving multi-linear systems. In this paper, we consider the Jacobi-type method which works better than the other two ones and apply a new preconditioner. The convergence of proposed preconditioned iterative method is studied. It is shown that the new approach is superior to the recently examined one in the literature. Numerical experiments illustrate the validity of theoretical results and the efficiency of the proposed preconditioner.
Algorithms, 2016
A homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.
Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs
Complexity, 2017
In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is 5m-11, where m is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In a...
On the preconditioned Jacobi method for solving large linear systems
Computing, 1982
On the Preconditioned Jacobi Method for Solving Large Linear Systems. This paper is concerned with the application of preconditioning techniques to the well known Jacobi iterative method for solving the finite difference equations derived from the discretization of self-adjoint elliptic partial differential equations. The convergence properties of this one parameter preconditioned method are analyzed and the value of the optimum preconditioning parameter and the performance of the method determined for a variety of standard problems.
Jacobi Operational Matrix and its Application for Solving Systems of ODEs
Differential Equations and Dynamical Systems, 2015
In this paper, we implement the shifted Jacobi operational matrix of derivative with spectral tau method and collocation method for numerical solution for the systems of linear and non-linear ordinary differential equations subject to initial or boundary conditions. By means of this approach, such problems are reduced for solving a system of algebraic equations and are greatly simplified the problems. We compare the obtained numerical results with the exact solutions. Also, we present and prove several theorems, which are related to the convergence of the proposed methods. Finally, some numerical test examples are presented to illustrate the validity and the great potential of the proposed technique.
Mathematical Methods in the Applied Sciences, 2015
In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high‐order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought‐for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.
Comparison results between Jacobi and other iterative methods
Journal of Computational and Applied Mathematics, 2004
Some comparison results between Jacobi iterative method with the modiÿed preconditioned simultaneous displacement (MPSD) iteration and other iterations, for solving nonsingular linear systems, are presented. It is showed that spectral radius of Jacobi iteration matrix B is less than that of several iteration matrices introduced in Liu (J. Numer. Methods Comput. Appl. 1 (1992) 58) under some conditions.
A Third Refinement of Jacobi Method for Solutions to System of Linear Equations
FUDMA JOURNAL OF SCIENCES
Solving linear systems of equations stands as one of the fundamental challenges in linear algebra, given their prevalence across various fields. The demand for an efficient and rapid method capable of addressing diverse linear systems remains evident. In scenarios involving large and sparse systems, iterative techniques come into play to deliver solutions. This research paper contributes by introducing a refinement to the existing Jacobi method, referred to as the "Third Refinement of Jacobi Method." This novel iterative approach exhibits its validity when applied to coefficient matrices exhibiting characteristics such as symmetry, positive definiteness, strict diagonal dominance, and -matrix properties. Importantly, the proposed method significantly reduces the spectral radius, thereby curtailing the number of iterations and substantially enhancing the rate of convergence. Numerical experiments were conducted to assess its performance against the original Jacobi method, t...