Finite difference method is the earliest method used in computer numerical simulation, and it is still widely used today. In this method, the solution domain is divided into difference grids, and the continuous solution domain is replaced by finite grid nodes. The finite difference method uses Taylor series expansion and other methods to discretize the derivative in the control equation by replacing the difference quotient of the function values on the grid nodes, thus establishing an algebraic equation with the values on the grid nodes as unknowns. This method is an approximate numerical solution, which directly turns the differential problem into an algebraic problem. Mathematical concepts are intuitive and simple to express. It is an early mature numerical 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, sub-region method belongs to the basic method of finite volume generation. 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.

The finite element method is based on variational principle and weighted residual method. 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 by means 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. In the finite element method, the computational domain is divided into finite non-overlapping and interconnected elements, and the basis function is selected in each element, and the true solution in the element is approximated by the linear combination of the basis functions of the elements. The whole basis function in the whole computational domain can be regarded as composed of the basis function of each element, and the solution in the whole computational domain can be regarded as composed of approximate solutions on all elements. Common finite element calculation methods include Ritz method, Galerkin method and least square method, which are developed from variational method and weighted residual method. According to the difference between weight function and interpolation function, finite element method is also divided into various calculation formats. There are collocation method, moment method, least square method and Galerkin method for the choice of weight function. For the weight function, Galerkin method takes the weight function as the basic function in the approximation function. The least square method makes the weight function equal to the residual itself, and the minimum value of the inner product is the minimum square error of the coefficient. In the configuration method, firstly, n configuration points are selected in the calculation domain. Let the approximate solution strictly satisfy the differential equation at the selected n configuration points, that is, let the equation margin be 0 at the configuration points.