1 Linear Equation Systems in the Numerical So
نویسنده
چکیده
1 Linear Equation Systems in the Numerical Solution of PDE’s 3 1.1 Examples of PDE’s . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Weak Formulation of Poisson’s Equation . . . . . . . . . . . . 6 1.3 Finite-Difference-Discretization of Poisson’s Equation . . . . . 7 1.4 FD Discretization for Convection-Diffusion . . . . . . . . . . 8 1.5 Irreducible and Diagonal Dominant Matrices . . . . . . . . . 9 1.6 FE (Finite Element) Discretization . . . . . . . . . . . . . . . 11 1.7 Discretization Error and Algebraic Error . . . . . . . . . . . . 14 1.8 Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9 Aim of a Multigrid Algorithm . . . . . . . . . . . . . . . . . . 15 1.10 Jacobi and Gauss-Seidel Iteration . . . . . . . . . . . . . . . . 15 1.10.1 Ideas of Both Methods . . . . . . . . . . . . . . . . . . 15 1.11 Convergence Rate of Jacobi and Gauss-Seidel Iteration . . . . 18 1.11.1 Analysis of the Convergence of the Jacobi Method . . 18 1.11.2 Iteration Method with Damping Parameter . . . . . . 20 1.11.3 Damped Jacobi Method . . . . . . . . . . . . . . . . . 20 1.11.4 Analysis of the Damped Jacobi method . . . . . . . . 20 1.11.5 Heuristic approach . . . . . . . . . . . . . . . . . . . . 22
منابع مشابه
P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrodinger equation
Many simulation algorithms (chemical reaction systems, differential systems arising from the modeling of transient behavior in the process industries and etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta technique are used. For the simulation of chemical procedures the ra...
متن کاملMinimal solution of fuzzy linear systems
In this paper, we use parametric form of fuzzy number and we converta fuzzy linear system to two linear system in crisp case. Conditions for the existence of a minimal solution to $mtimes n$ fuzzy linear equation systems are derived and a numerical procedure for calculating the minimal solution is designed. Numerical examples are presented to illustrate the proposed method.
متن کاملA New Two-stage Iterative Method for Linear Systems and Its Application in Solving Poisson's Equation
In the current study we investigate the two-stage iterative method for solving linear systems. Our new results shows which splitting generates convergence fast in iterative methods. Finally, we solve the Poisson-Block tridiagonal matrix from Poisson's equation which arises in mechanical engineering and theoretical physics. Numerical computations are presented based on a particular linear system...
متن کاملCAS WAVELET METHOD FOR THE NUMERICAL SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH LOGARITHMIC SINGULAR KERNELS
In this paper, we present a computational method for solving boundary integral equations with loga-rithmic singular kernels which occur as reformulations of a boundary value problem for the Laplacian equation. Themethod is based on the use of the Galerkin method with CAS wavelets constructed on the unit interval as basis.This approach utilizes the non-uniform Gauss-Legendre quadrature rule for ...
متن کاملNumerical Solution of Heun Equation Via Linear Stochastic Differential Equation
In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreo...
متن کاملNumerical solution of fuzzy linear Fredholm integro-differential equation by \fuzzy neural network
In this paper, a novel hybrid method based on learning algorithmof fuzzy neural network and Newton-Cotesmethods with positive coefficient for the solution of linear Fredholm integro-differential equation of the second kindwith fuzzy initial value is presented. Here neural network isconsidered as a part of large field called neural computing orsoft computing. We propose alearning algorithm from ...
متن کامل