, the 2-dimensional Poisson equation can be written in a matrix form Considering that the boundary values are known, we arrange the remaining nodes in columns to form a new column vector. Where the corresponding truncation error is , then for the problem we have the following algorithm, Similarly, in y direction, the points inīoth in x and y directions we use the central difference method respectively, as shown in Figure 1, we haveįigure 1. Points and give an equidistant discretization, that is The 2-dimensional Poisson equation we will discuss is as follows,įirst, we will give a discretization in x and y directions. The finite difference method with five points will be applied first, then the Chebyshev spectral method will be considered. In this part, we will show two different schemes to solve the 2-dimensional Poisson equation. Numerical Algorithms for 2-Dimensional Poisson Equation For a more detailed discussion about the properties of Chebyshev and other polynomials, one can refer to. There are some other polynomials such as Legendre polynomials, Jacobi polynomials, etc. The Chebyshev polynomials satisfy the following orthogonal properties with weightįirst-order derivative of Chebyshev polynomials This feature of the Chebyshev points can help one overcome Runge phenomenon in interpolation. These Chebyshev points are highly clustered on both sides of the interval. Ī big difference between Chebyshev polynomials and other polynomials is that Chebyshev polynomials have an explicit relation, i.e.Īre called Chebyshev points. , the polynomials with the following relations are called Chebyshev polynomials. The Chebyshev points will be used to construct the differentiation matrices, which is the key point to obtain high-precision solutions. Some basic contents including Chebyshev polynomials and Chebyshev points will be introduced in this part. In this work, we will use Chebyshev spectral method to find the numerical solution of 2-dimensional Poisson equation. The spectral method is a well-developed algorithm, which has infinite order accuracy and exponential order convergence speed in theory. For the in-depth discussion of these methods, we can refer to. In addition to the finite difference method and spectral method, the finite element method is also an effective method to deal with differential equations. Especially for multidimensional problems, such as 2-dimensional or 3-dimensional Poisson equation if we use the finite difference method, we need to calculate so many nodes the amount of calculation is very large. Compared to the finite difference method the spectral method has high accuracy and less computation. But if we want to continue to improve the accuracy of the algorithm based on the 9-point difference method, it will be very difficult. The error of the 9-point difference scheme is fourth-order. In order to improve the accuracy, some scholars proposed a new method based on 5-point difference scheme, such as 9-point difference scheme. The precision of the traditional 5-point difference method is not good enough. For the second-order parabolic type differential equation, the finite difference method is a popular discretization method. Therefore, the numerical algorithm is a good way to deal with this problem. In general, the Poisson equation is hard to get the analytical solution, only a few can find the exact solution. The above Equation (1) is the 2-dimensional Poisson equation, where In the following part, we will focus on Poisson equation with the Dirichlet boundary condition. In recent several years, some researchers find that the 2-dimensional Poisson equation with Dirichlet boundary condition is a good tool to cope with seamless image composite problems. Now Poisson equation has been applied to modelling the temperature distribution of stable temperature field with a stable heat source or without internal heat source, the stable non-rotating flow of incompressible fluid in hydrodynamics, etc. Poisson pointed out that if the density of gravitational field is considered, the Laplace equation should have a new form i.e. Laplace discussed the problem of the gravitational field. Can be dated back to 1782 when the French mathematician P.S.
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