Numerical Computing For PDEs And The Study Of Incremental Unknowns

In this paper, based on the finite difference discretization of partial differential equations and the advantage that the incremental unknowns(IU) on uniform meshes can reducing the condition number of coefficient matrix effectively, we study the heat equations with the time-dependent coefficients in the 1 and 2 dimensions and a class of generalized three dimensional convection-diffusion equations with this method. The effectiveness of this method are established by the theoretical analysis and the numerical results. But for many problems, especially the boundary layer problems, nonuniform meshes are more flexible and efficient than the uniform meshes. So, the discretization technique and then the incremental unknowns on nonuniform meshes(NIU) become more and more important. With the Dirichlet problem, we theoretically and numerically analyze the condition number of the coefficient matrix with NIU in dimension 1 and 2.For the one dimensional heat equation with time-dependent coefficient, we propose a kind of IU-type semi-implicitθ-schemes and carefully study the stability, error estimation and condition number of these schemes. The theoretical analysis shows that a better stability condition was obtained when 9 close to 1/2. For the two dimensional case, we construct an alternating direction IU-type semi-implicit scheme. The stability condition of this new scheme is obtained with the Fourier method. Numerical results show that this new scheme is more efficient than the classical alternating directional scheme for some problems when r satisfies the stability condition.With the finite difference discretization techniques and the IU method, we get a nonsymmetric and positive-definite linear system when considering a class of generalized three dimensional convection-diffusion equations. Considering that the condition number of this coefficient matrix is much better than the matrix without IU, we use this method in conjunction with several classical iterative methods to approximate the solution of the system. After estimating the condition number of IU-type coefficient matrix, we numerically confirm that these IU-type iterative methods are much more efficient.Note that the finite difference discretization techniques and the IU method are defined on the uniform meshes. But for many problems, for example, the boundary layer or hydromechanics problems, the methods defined on the uniform mesh are no longer work very well. Hence the NIU method becomes more and more important. With the question that does the NIU method also can reduce the condition number of coefficient matrix as the IU method, we give a carefully analysis both in 1 and 2 dimensions. The theoretical results show that NIU method also work effectively, which was established by the numerical results.