86,360 research outputs found
A Parameterized multi-step Newton method for solving systems of nonlinear equations
We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft
Numerical solution of a coupled pair of elliptic equations from solid state electronics
Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem
Fast and Efficient Numerical Methods for an Extended Black-Scholes Model
An efficient linear solver plays an important role while solving partial
differential equations (PDEs) and partial integro-differential equations
(PIDEs) type mathematical models. In most cases, the efficiency depends on the
stability and accuracy of the numerical scheme considered. In this article we
consider a PIDE that arises in option pricing theory (financial problems) as
well as in various scientific modeling and deal with two different topics. In
the first part of the article, we study several iterative techniques
(preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis
have been used to design various preconditioners to improve the convergence
criteria of iterative solvers. We implement a multigrid (MG) iterative method.
In fact, we approximate the problem using a finite difference scheme, then
implement a few preconditioned Krylov subspace methods as well as a MG method
to speed up the computation. Then, in the second part in this study, we analyze
the stability and the accuracy of two different one step schemes to approximate
the model.Comment: 29 pages; 10 figure
Solving 2D Time-Fractional Diffusion Equations by Preconditioned Fractional EDG Method
Fractional differential equations play a significant role in science and technology given that several scientific problems in mathematics, physics, engineering and chemistry can be resolved using fractional partial differential equations in terms of space and/or time fractional derivative. Because of new developments in the analysis and understanding of many complex systems in engineering and sciences, it has been observed that several phenomena are more realistically and accurately described by differential equations of fractional order. Fast computational methods for solving fractional partial differential equations using finite difference schemes derived from skewed (rotated) difference operators have been extensively investigated over the years. The main aim of this paper is to examine a new fractional group iterative method which is called Preconditioned Fractional Explicit Decoupled Group (PFEDG) method in solving 2D time-fractional diffusion equations. Numerical experiments and comparison with other existing methods are given to confirm the superiority of our proposed method
Iterative Methods for Problems in Computational Fluid Dynamics
We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible Navier-Stokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convection-diffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the Navier-Stokes equations
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