1,720,976 research outputs found
Monotone iterative methods for solving nonlinear partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
Get PDFA key aspect of the simulation process is the formulation of proper mathematical models. The model must be able to emulate the physical phenomena under investigation. Partial differential equations play a major role in the modelling of many processes which arise in physics, chemistry and engineering. Most of these partial differential equations cannot be solved analytically and classical numerical methods are not always applicable. Thus, efficient and stable numerical approaches are needed. A fruitful method for solving the nonlinear difference schemes, which discretize the continuous problems, is the method of upper and lower solutions and its associated monotone iterations. By using upper and lower solutions as two initial iterations, one can construct two monotone sequences which converge monotonically from above and below to a solution of the problem. This monotone property ensures the theorem on existence and uniqueness of a solution. This method can be applied to a wide number of applied problems such as the enzyme-substrate reaction diffusion models, the chemical reactor models, the logistic model, the reactor dynamics of gasses, the Volterra-Lotka competition models in ecology and the Belousov-Zhabotinskii reaction diffusion models.
In this thesis, for solving coupled systems of elliptic and parabolic equations with quasi-monotone reaction functions, we construct and investigate block monotone iterative methods incorporated with Jacobi and Gauss--Seidel methods, based on the method of upper and lower solutions. The idea of these methods is the decomposition technique which reduces a computational domain into a series of nonoverlapping one dimensional intervals by slicing the domain into a finite number of thin strips, and then solving a two-point boundary-value problem for each strip by a standard computational method such as the Thomas algorithm. We construct block monotone Jacobi and Gauss-Seidel iterative methods with quasi-monotone reaction functions and investigate their monotone properties. We prove theorems on existence and uniqueness of a solution, based on the monotone properties of iterative sequences. Comparison theorems on the rate of convergence for the block Jacobi and Gauss-Seidel methods are presented. We prove that the numerical solutions converge to the unique solutions of the corresponding continuous problems. We estimate the errors between the numerical and exact solutions of the nonlinear difference schemes, and the errors between the numerical solutions and the exact solutions of the corresponding continuous problems. The methods of construction of initial upper and lower solutions to start the block monotone iterative methods are given
Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem
No full textAbstractThis paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic reaction–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone domain decomposition algorithm is estimated. Numerical experiments are presented
Monotone Schwarz iterates for a semilinear parabolic convection–diffusion problem
No full textAbstractThis paper deals with a discrete monotone iterative algorithm for solving a nonlinear singularly perturbed convection–diffusion problem of parabolic type. On each time level, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear upwind difference scheme obtained after discretisation of the continuous problem. A monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. The rate of convergence of the monotone Schwarz method is estimated. Uniform convergence properties of the monotone domain decomposition algorithm are studied. Numerical experiments are presented
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Uniform convergent methods on arbitrary meshes for singularly perturbed problems with piecewise smooth coefficients
Get PDFThis paper deals with uniform convergent methods for solving singularly perturbed two-point
boundary value problems with piecewise smooth coefficients. Construction of the numerical methods
is based on locally exact schemes or on local Green’s functions. Uniform convergent properties
of the proposed methods on arbitrary meshes are proven. Numerical experiments are presented
A monotone domain decomposition algorithm for nonlinear parabolic difference schemes
Get PDFA monotone domain decomposition algorithm for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear reaction-diffusion problem of parabolic type, is presented and is shown to converge monotonically either from above or from below to a solution of the system. The algorithm is based on a modification of the Schwarz alternating method and the method of upper and lower solutions. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the system, and is potentially parallelisable. Numerical experiments for a model problem from chemical engineering are presented
Inexact monotone methods for solving nonlinear elliptic problems
Get PDFWe numerically solving semilinear elliptic problems with the method of upper and lower solutions. Inexact monotone iterative methods are constructed, where monotone linear systems are solved by the Jacobi or Gauss--Seidel methods only approximately. The inexact monotone methods combine the quadratic monotone iterative method at outer iterations and the Jacobi or Gauss--Seidel methods at inner iterations, and possess global monotone convergence. Results of numerical experiments are presented.
References B. Abraham and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979. doi:10.1137/1.9781611971262 I. Boglaev, Uniform quadratic convergence of monotone iterates for semilinear singularly perturbed elliptic problems. Lecture Notes in Computational Science and Engineering 81:37–46, 2011. doi:10.1007/978-3-642-19665-2_5 I. Boglaev, Monotone relaxation iterates and applications to semilinear singularly perturbed problems. Int. J. Numer. Anal. Mod.(B) 2:402–414, 2011. http://www.math.ualberta.ca/ijnamb/Volume-2-2011/No-4-11/2011-04-08.pdf I. Boglaev, An inexact monotone method for solving semilinear parabolic problems. Appl. Math. Comput. 219:3253–3263, 2012. doi:10.1016/j.amc.2012.09.067 R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods. SIAM J. Numer. Anal. 19:400–408, 1982 doi:10.1137/0719025 S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17:16–32, 1996. doi:10.1137/0917003 P. Knabner and L. Angerman, Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, New York, 2003. doi:10.1007/b97419 C. V. Pao, Nonlinear Parabolic and Elliptic Equations. Springer, New York, 1992. doi:10.1007/978-1-4615-3034-3 C. V. Pao, Accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems. Computers Math. Applic., 46:1535–1544, 2003. doi:10.1016/S0898-1221(03)00381-
Monotone iterates for solving systems of semilinear elliptic equations and applications
Get PDFConsider monotone finite difference iterative algorithms for solving coupled systems of semilinear elliptic equations. A monotone domain decomposition algorithm based on a modification of Schwarz alternating method and on decomposition of a computational domain into nonoverlapping subdomains is constructed. Advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the nonlinear discrete problem, and is potentially parallelisable. The monotone domain decomposition algorithm is applied to a gas-liquid interaction model. Numerical experiments confirm theoretical results
A block monotone domain decomposition algorithm for a semilinear convection–diffusion problem
No full textAbstractThis paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection–diffusion problem. A block monotone domain decomposition algorithm based on a Schwarz alternating method and on block iterative scheme is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the nonlinear problem. The rate of convergence of the block monotone domain decomposition algorithm is estimated. Numerical experiments are presented
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