Improved finite integration method for partial differential equations

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Abstract

Based on the recently developed finite integration method (FIM) for solving one-dimensional partial differential equations by using the trapezoidal rule for numerical quadrature, we improve in this paper the FIM with an alternative extended Simpson׳s rule in which the Cotes and Lagrange formulas are used to determine the first order integral matrix. The improved one-dimensional FIM is then further extended to solve two-dimensional problems. Numerical comparison with the finite difference method and the FIM (Trapezoidal rule) are performed by several one- and two-dimensional real application including the Poisson type differential equations and plate bending problems. It has been shown that the newly revised FIM has made significant improvement in terms of accuracy compare without much sacrifice on the stability and efficiency.

Introduction

Partial differential equations (PDEs) commonly appear in mathematical modeling to describe a wide variety of physical phenomena such as fluid and solid mechanics. The properties and characteristics of the physical phenomena can then be understood from the closed form solutions to these PDEs. However, under various boundary conditions and real problem configuration, it is very rare that these models can be solved in closed form solutions. Due to the advancement of computational methods, numerical approximations can usually be achieved inexpensively to give high accuracy together with a reliable bound on the error between the analytical solution and its numerical approximation. There are many numerical techniques available for solving differential equations [1], [2], [3], [4], [5] among which the finite difference method (FDM), Finite Element Method (FEM) and Boundary Element Method (BEM) are commonly used.

Recently, Wen et al. [6] and Li et al. [7], [8] developed a new finite integration method (FIM) for solving one- and two-dimensional partial differential equations and successfully demonstrated its applicability for solving nonlocal elasticity problems. It has been shown that the FIM gives a much higher degree of accuracy than the FDM and the Point Collocation Method (PCM). In this paper, an improved FIM is developed by using an alternative extended Simpson׳s rule, Cotes integral formula, and Lagrange formula for solving one- and two-dimensional partial differential equations. Similar to the FDM and the PCM, a finite number of points, known as field points, are distributed in the computational domain. The field points are generated either uniformly (grid) along the independent coordinate or randomly distributed in the domain. The integration matrix of the first order is obtained by direct integration with Simpson׳s rules, Cotes formula, and Lagrange formula. Based on these first order integration matrices, the multi-layer finite integration matrix can easily be obtained. To demonstrate the accuracy and efficiency of the improved FIM, several one-dimensional and two-dimensional numerical examples are given and compared with the FDM and analytical solution.

Section snippets

Trapezoidal rule (TR)

A simple computational scheme for integration was introduced in [6], [7], which was called an Ordinary Linear Approach (OLA) as follows. LetU(xk)=0xku(ξ)dξi=1kakiu(xi).

For the most simple trapezoidal rule, the coefficients area1i=0,aki={0.5h,i=1,hi=2,3,...,k1,0.5hi=k,0i>k,where xi=(i1)h,h=b/(N1),i=1,2,...,N are nodal points in [0, b], and x1=0,xN=b. Note that (1) can be written in matrix form asU=Auwhere U=[U1,U2,...,UN]T, u=[u1,u2,...,uN]T,A=(aki)=h(0000001/21/200001/211/20001/2111/200...

Application of FIM to one-dimension problems

Example 1

Ordinary differential equation

To compare the difference between the FIM (trapezoidal rule) and FIM (extended Simpson׳s rules), we consider the following ordinary differential equation [6]d2udx22(1+2x2)u=0,x[0,1],where u(0)=1and u׳(0)=0. Applying integration operation to (28) and using the notations derived in Section 2.1, we haveuABu=c0I,where c0 is an arbitrary integral constant, I is unit diagonal matrix, andB=(2(1+2x1)0...002(1+2x2)...0............00...2(1+2xN)).Applying the integration

Improved FIM for two-dimensional problems

For two-dimensional problems, consider a uniform distribution of grid points as shown in Fig. 2. The integration matrix is defined asUx(x,y)=0xu(ξ,y)dξ,Ux(xk,yk)=0xku(ξ,yk)dξ,and the total number of points is k=N1(j1)+i, where i and j denote the total number of columns and the total number of rows, respectively. This numbering system is called the global number system. We can then express each nodal value of integration (40) in a matrix form asUx=Axu,where the integral nodal value Ux=[Ux1,Ux2

Application of the FIM to two-dimensional problems

Example 3

Two dimensional potential problem

We first consider a quarter of a cylinder as shown in Fig. 3(a). The partial differential equation in the polar coordinate system is2ur2+1rur+1r22uθ2=2r2,1r2,0θπ/2,with boundary condition u(r,0)=u(r,π/2)=0,u(1,θ)=1/4sin22θ,u(2,θ)=4sin22θ. The analytical solution is given by u(r,θ)=r4/4sin22θ. Let x=r−1 and y=πθ/2, (55) in the polar coordinate system can be transformed to the following equation in the Cartesian coordinate system2ux2+x(u1+x)+u(1+x)2+4

Conclusion

In this paper, an improved FIM using three numerical quadrature formula, namely alternative extended Simpson׳s rule, Cotes integral, and Lagrange interpolation, are derived for solving both one-dimensional and two-dimensional problems. To solve the ordinary and partial differential equations, the different order of integration matrices are obtained by the integration matrix of the first order integration matrix. For two-dimensional problems, there are several arbitrary integral functions to be

Acknowledgements

The work of this paper was partially supported by a grant from the Research Council of Hong Kong Special Administration Region (Project No. CityU 101112) and the National Natural Science Foundation of China (Grant No. 11401422).

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