difference method
Numerical solutions of differential equations and integro-differential equations. The basic idea is to replace the continuous definite solution area with a grid composed of finite discrete points, which are called nodes of the grid; The function of continuous variables in the continuous definite solution region is approximated by the discrete variable function defined on the grid; The Wechat business approximation of the original equation and the definite solution condition, the integral is approximated by the integral sum, so the original differential equation and the definite solution condition are replaced by algebraic equations, that is, finite difference equations. By solving the equations, the approximate solution of the original problem at discrete points can be obtained. Then the approximate solution of the definite solution problem in the whole region is obtained from the discrete solution by interpolation method.
The steps of solving partial differential equations by finite difference method are as follows:
1, domain discretization, that is, subdividing the solution region of a given partial differential equation into a grid composed of finite lattice points;
2. Approximate substitution, that is, the derivative of each lattice point is replaced by finite difference formula;
3. Approximate solution. In other words, this process can be regarded as a process of replacing the solution of partial differential equation with an interpolation polynomial and its differential.
How to mesh the definite solution area according to the characteristics of the problem; How to discretize the original differential equation into difference equation and how to solve this algebraic equation group. In addition, in order to ensure the feasibility of the calculation process and the correctness of the calculation results, it is necessary to analyze the behavior of the difference equation theoretically, including the uniqueness and existence of the solution and the compatibility, convergence and stability of the difference scheme. For all kinds of difference schemes established by a differential equation, to have practical significance, a basic requirement is that they can arbitrarily approximate the differential equation, which is the compatibility requirement. In addition, whether the difference scheme is useful ultimately depends on whether the exact solution of the difference equation can arbitrarily approximate the solution of the differential equation, which is the concept of convergence. In addition, there is an important concept that must be considered, that is, the stability of difference schemes. Because the calculation process of the difference scheme is advanced layer by layer, the approximation of n+ 1 layer should be used until it is related to the initial value. If there are rounding errors in the previous layers, it will inevitably affect the values in the following layers. If the influence of errors is so great that the exact solution of the difference scheme is completely covered up, the scheme is unstable. On the contrary, if the error propagation can be controlled, the scheme is considered to be stable. Only in this case, the approximate solution of the difference scheme can arbitrarily approximate the exact solution of the difference equation in practical calculation. There are usually three ways to construct difference schemes. The most commonly used method is numerical differentiation, such as using difference quotient instead of WeChat service. Another method is called integral interpolation, because the differential equations obtained in practical problems often reflect some conservation principle in physics and can generally be expressed in integral form. In addition, the undetermined coefficient method can also be used to construct some high-precision difference schemes.
Finite volume method
The finite volume method is also called the controlled volume method.
The basic idea is: divide the calculation area into a series of non-repetitive control bodies, and make a control body around each grid point; By integrating the differential equation to be solved for each control volume, a set of discrete equations is obtained. The unknown is the value of the dependent variable at the grid point. In order to calculate the integral of the control volume, it is necessary to assume the changing law of the values between grid points, that is, the distribution outline of the piecewise distribution of the assumed values. Judging from the selection method of integral region, the finite volume method belongs to the sub-region method in the weighted residual method; From the approximate method of unknown solution, the finite volume method belongs to the discrete method of local approximation. In a word, the partition method belongs to the basic method of finite volume method. The basic idea of finite volume method is easy to understand and can be directly explained physically. The physical meaning of discrete equation is the conservation principle of dependent variable in finite control volume, just as differential equation represents the conservation principle of dependent variable in infinite control volume. The discrete equation obtained by finite volume method requires the integral conservation of dependent variables for any group of control volumes, and of course it is also true for the whole calculation area. This is an attractive advantage of the finite volume method. Some discrete methods, such as finite difference method, only when the grid is extremely fine can the discrete equation satisfy the integral conservation; However, even in the case of coarse mesh, the finite volume method shows accurate integral conservation. As far as discrete method is concerned, finite volume method can be regarded as the intermediate between finite element method and finite difference method. The finite element method must assume the variation law of values between grid points (that is, interpolation function) and take it as an approximate solution. The finite difference method only considers the values on grid points, and does not consider how the values between grid points change. The finite volume method only finds the node value, which is similar to the finite difference method; The finite volume method must assume the distribution of values between grid points when solving the control volume integral, which is similar to the finite element method.
Finite element method
Finite element method is an effective method to solve mathematical problems. Its basis is variational principle and weighted residual method, and its basic solution idea is to divide the calculation domain into finite non-overlapping units, select some suitable nodes in each unit as interpolation points for solving the function, rewrite the variables in the differential equation into linear expressions composed of the node values of each variable or its derivative and the selected interpolation function, and solve the differential equation discretely with the help of variational principle or weighted residual method. Different weight functions and interpolation functions are used to form different finite element methods. The finite element method was first applied to structural mechanics, and then it was gradually used to simulate fluid mechanics with the development of computer.
boundary element method
Boundary element method is a numerical method developed after finite element method. As early as 1970s, this method was adopted by the Department of Civil Engineering of Southampton University in the UK. C.A.Brebbia of our department strongly advocates the boundary element method internationally. Now this term has been widely accepted by scientists, and the boundary element method has been gradually applied to various fields.
The boundary element method transforms the partial differential equation of the studied problem into the boundary integral equation defined on the boundary, and then discretizes the boundary integral equation into algebraic equations containing only the unknown quantities of the boundary nodes. By solving this set of equations, the unknown quantity at the boundary node can be obtained, and then the unknown quantity in the research area can be obtained. It can not only deal with most problems that the finite element method is applicable to, but also deal with infinite problems that the finite element method is not easy to solve.
Because the boundary element method only divides the elements on the boundary of the research area, the dimension of the solution is reduced: the three-dimensional problem becomes a two-dimensional problem, and the two-dimensional problem becomes a one-dimensional problem. Solving a problem requires a small set of equations, which is beneficial to saving memory and computing time. In addition, because the boundary element method introduces the basic solution, it has the characteristics of combining analysis with discretization, so it has high accuracy.
Spline boundary element method
Spline boundary element method has the advantages of symmetry, positive definiteness, sparsity of coefficient matrix and ignoring natural boundary conditions. , and it has its own unique advantages, high accuracy, less calculation, is an efficient calculation method. Its disadvantage is poor universality, and it is only applicable to some special shapes and boundary conditions composed of several rectangles.
Finite analysis method
Finite analysis method is an improvement on the basis of finite element method, which was put forward by Chinese American Chen in 1970s. The differential equation is linearized, and the analytical solution of the differential equation is obtained on the local element under the condition of interpolating the approximate boundary, forming the whole linear algebraic equations. Finite analysis method is a combination of analytical method and numerical method, and it is a progress of computational fluid dynamics. Its advantages are high calculation accuracy, automatic windward characteristic, good calculation stability and fast convergence speed, but the element coefficient contains complex infinite series, which brings some difficulties to practical calculation and theoretical analysis. In recent years, Well Lee and others put forward the mixed finite analysis method, which introduced the idea of finite difference, avoided the calculation of infinite series, and greatly improved the application value of this method. However, both finite analysis method and mixed finite analysis method have some shortcomings, such as complex finite analysis coefficient and slow calculation speed.
Numerical integral transformation method
Numerical integral transformation is a hybrid method of numerical solution and analytical solution based on the principle of universal integral transformation. Its basic idea is to decompose the original problem into an eigenvalue problem and a reduced-dimension definite solution problem. For the simple eigenvalue problem, a closed analytical expression can be obtained by analytical method, while the definite solution problem is still solved by numerical method. However, because the definite solution problem reduces the dimension and independent variables compared with the original problem, it is easy to solve. This method has the characteristics of both analytical solution and numerical solution. Combining the two solutions organically, we only need to give a numerical solution on a coordinate (space or time) variable, and through the linear combination of analytical solution and numerical solution, we can get the value of a specific point in the region, which greatly reduces the calculation workload.