Pde Numerical Solution Of Linear Algebraic Equations To Solve

We often use the finite difference method, finite element method and the finite volume method and other methods in the numerical solution of partial differential equations.The essence of the problem is to discretize continuous ones.In the end,it always comes down to solving equations.While the large linear equations are common, and nonlinear equations are often converted into linear algebraic equations to solve. So for the numerical solution of partial differential equations, solving algebraic equations plays a very important part.Solving equations is mainly to numerical algebra areas. Taking the numerical solution of partial differential equations as their research direction, the scientific and technological personnels often do not go to study methods of solving equations and theoretical analysis. Actually they pay attention to the use of the algorithm, the conclusion of the algorithm and the effects of the algorithm (the calculation accuracy, speed, etc). The purpose of this paper is to discuss the various methods of large line algebraic quations,combining theories with practice, and providing elections of algorithm when solving equations. In the rounding error analysis part one example of calculating failure due to rounding error is gived and some places needed to perfect of theory of rounding error. The Poisson equations problem for typical numerical solution of the partial differential equations, further experiment is did. Knowing continuous problems after finite difference scheme discretization of large sparse type equations produced by the direct method to the process of programming and the final computation effects. Finally we discussed one-dimensional two-point boundary value problems with numerical experiments of tridiagonal equation of chase-after method.In the direct method part, it should be pointed out that, the size of condition number in relationship with matrix morbid good state, the theoretical analysis method of judging whether a specific problem is morbid or good state are needed much more research. This paper discussed the constructions of some places which are not clear or confusingly explained in numerical analysis materials. And we put forward some questions. Secondly for Gauss column primary elimination method we do the the analysis of combining specific theory with practice:we do the numerical experiments about dense matrix.Combining theory with practice we give the answer to equations type, the scale, the calculation speed equations, rounding error theory and rounding errors due to affecting the final results of calculation accuracy problems; In the iterative method part, we analyze the convergent conditions and convergent rate of iteration method, and we combine with the common problem to do the synthesis of various algorithms for numerical comparison of partial differential equation model experiments. Among them,as the Jacobi method for the problem of slow convergence, this thesis prestents Jacobi combining Chebyshev semi-iterative algorithm to the five-point difference scheme of second order Poisson equation, and the accuracy is verified through numerical experiments.As the optimum relaxation factor of successive overrelaxation (SOR) iterative method is difficult to find in the theory, the thesis also presents nonlinear least squares rectangular area of data under different partition to five-point difference scheme of second-order Poisson equation. It comes to the optimum relaxation factor of the rational regression formula, and its accuracy is verified through numerical experiments. Finally, numerical experiments of the conjugate gradient method (CG method), Jacobi iteration, Gauss-Seidel iteration and SOR iterative method are compared.