Computer methods for numerical simulation

The finite-difference method (FDM) is the earliest method used in computer numerical simulation and is still widely used today. The method divides the solution domain into differential grids, replacing the continuous solution domain with a finite number of grid nodes. The finite difference method uses methods such as Taylor series expansion to discretize the derivatives of the control equations by replacing them with the quotients of the differences of the function values at the grid nodes, so as to build a system of algebraic equations in which the values at the grid nodes are the unknowns. This method is a kind of approximate numerical solution that directly turns the differential problem into an algebraic problem, with intuitive mathematical concepts and simple expression, which is a relatively mature numerical method developed earlier.　For the finite difference format, there are first-order format, second-order format and high-order format in terms of the accuracy of the format. Considering the spatial form of the difference, it can be divided into center format and upwind format. Considering the influence of the time factor, the difference format can also be divided into explicit format, implicit format, and explicit-implicit alternating format. At present, the common differential formats are mainly combinations of the above forms, and different combinations constitute different differential formats. Differential methods are mainly applied to structured meshes, and the step size of the mesh is usually decided according to the actual terrain and the Crown stabilization conditions.

There are many ways to construct the difference, and the Taylor series expansion method is the main one used at present. There are three basic differential expressions: first-order forward differential, first-order backward differential, first-order center differential and second-order center differential, of which the first two formats are for the first-order computational accuracy, and the last two formats are for the second-order computational accuracy. By combining these different differential formats in time and space, different differential calculation formats can be combined.

The finite element method is based on the variational principle and the weighted margin method, and its basic idea is to divide the computational domain into a finite number of non-overlapping units, and in each unit, select some suitable nodes as the interpolation points of the solution function, rewrite the variables in the differential equation as linear expressions composed of the node values of the variables or their derivatives and the interpolation function selected, and then, with the help of the variational principle or the weighted margin method, the differential equation can be calculated by the variational principle or the weighted margin method, and the interpolation function can be calculated by the variational principle. The differential equations are solved discretely by means of the variational principle or the weighted residual method. Different forms of weighting and interpolation functions are used to form different finite element methods. The finite element method was first applied to structural mechanics, and then slowly used for numerical simulation of fluid mechanics with the development of computers. In the finite element method, the computational domain is discretized into a finite number of non-overlapping and interconnected units, the basis function is selected in each unit, and the linear combination of the unit basis functions is used to approximate the true solution in the unit, the overall basis function on the entire computational domain can be viewed as the basis function of each unit is composed of the entire computational domain can be viewed as an approximation of the solution by all the units on the composition of the solution. In the numerical simulation of river channels, the common finite element calculation methods are the Ritz method and Galyokin method developed from the variational method and the weighted residual method, and the least squares method. The finite element methods are also categorized into a variety of computational formats depending on the weighting function and interpolation function used. From the selection of the weight function, there are the configuration method, the method of moments, the method of least squares and Galyokin's method, from the shape of the computational cell mesh, there are triangular mesh, quadrilateral mesh and polygonal mesh, and from the interpolation function of the accuracy of the division, it is divided into the linear interpolation function and the higher interpolation function, etc. Different combinations also constitute different finite element calculation formats. Different combinations also constitute different finite element calculation formats. For the weight function, the Galerkin method is to take the weight function as the basis function in the approximation function; the least squares method is to make the weight function equal to the residual itself, and the minimum value of the inner product is to minimize the square error of the coefficients of the surrogate; in the configuration method, the first in the computational domain to select the N configuration points. So that the approximate solution in the selected N configuration points strictly satisfy the differential equation, that is, in the configuration points so that the equation margin of 0. Interpolation function generally consists of different powers of polynomials, but also used trigonometric or exponential functions composed of the product of representation, but the most commonly used polynomial interpolation function. Finite element interpolation function is divided into two categories, one only requires the interpolating polynomial itself at the interpolation point to take a known value, known as the Lagrange (Lagrange) polynomial interpolation; the other not only requires the interpolating polynomial itself, but also requires it to take a known value of the derivative value of the interpolating point, known as the Hammett (Hermite) polynomial interpolation. Unitary coordinates have Cartesian Cartesian coordinate system and factorless natural coordinates, symmetric and asymmetric, etc.. The commonly used factorless coordinates are a localized coordinate system whose definition depends on the geometry of the cell and is viewed as a length ratio in one dimension, an area ratio in two dimensions, and a volume ratio in three dimensions. In two-dimensional finite elements, triangular cells were the first to be applied, and recently the use of quadrilateral and other parametric elements has become more widespread. For two-dimensional triangular and quadrilateral power unit, often used interpolation function for the Lagrange interpolation linear interpolation function in the rectangular coordinate system and second or higher order interpolation function, linear interpolation function in the area coordinate system, second or higher order interpolation function.

For the finite element method, its basic ideas and solution steps can be summarized as

(1) the establishment of integral equations, according to the principle of variational or equation residual and weight function orthogonalization principle, the establishment of differential equations and differential equations with the first edge of the value of the problem equivalent to the integral expression, which is the starting point of the finite element method.

(2) Regional unit division, according to the shape of the solution region and the physical characteristics of the actual problem, the region is divided into a number of interconnected, non-overlapping units. Regional unit division is the use of finite element method of the preliminary preparatory work, this part of the workload is relatively large, in addition to the computational unit and node numbering and determine the relationship between each other, but also to indicate the location of the node coordinates, but also need to list the natural boundary and the essential boundary of the node serial number and the corresponding boundary value.

(3) determine the unit basis function, according to the number of nodes in the unit and the requirements of the accuracy of the approximate solution, select the interpolation function to meet certain interpolation conditions as the unit basis function. Finite element method in the base function is selected in the unit, because the unit has a regular geometry, in the selection of the base function can follow certain laws.

(4) unit analysis: the solution function of each unit with a linear combination of unit basis function expression for the approximation; then the approximation function into the integral equation, and the integration of the unit region, can be obtained with the coefficients to be determined (that is, the nodes in the unit of the parameter values) of the set of algebraic equations, known as the unit of finite element equations.

(5) Overall synthesis: After obtaining the unit finite element equation, all the unit finite element equations in the region are summed up according to a certain law to form the overall finite element equation.

(6) Boundary conditions: Generally, there are three forms of boundary conditions, which are divided into essential boundary conditions (Dirichlet boundary conditions), natural boundary conditions (Riemann boundary conditions), and mixed boundary conditions (Cauchy boundary conditions). For the natural boundary condition, it can be satisfied automatically in the integral expression. For the essential boundary conditions and hybrid boundary conditions, the overall finite element equation should be modified according to certain rules.

(7) Solve the finite element equations: the overall finite element equations corrected according to the boundary conditions are a closed set of equations containing all the unknown quantities to be determined, and can be solved by appropriate numerical methods to obtain the function value of each node.

The Finite Volume Method is also known as the Control Volume Method. The basic idea is: the calculation area is divided into a series of non-repeating control volume, and so that each grid point around a control volume; the differential equation to be solved for each of the control volume integral, a set of discrete equations. The unknowns in these are the values of the dependent variables at the grid points. In order to find the integral of the control volume, it is necessary to assume a pattern of change of values between grid points, i.e., a distribution profile that assumes the distribution of segments of values. From the point of view of the selection method of the integration region, the finite volume method belongs to the subregion method in the weighted residual method; from the point of view of the approximation method of the unknown solution, the finite volume method belongs to the discretization method using local approximation. In short, the subregion method belongs to the basic method of finite volume hair.

The basic idea of the finite volume method is easy to understand and leads to a direct physical interpretation. The physical meaning of the discrete equations is the principle of conservation of the dependent variable in a control volume of finite size, just as the differential equations represent the principle of conservation of the dependent variable in an infinitely small control volume. The discrete equations derived by the finite volume method require that the conservation of the integral of the dependent variable be satisfied for any set of control volumes, and naturally for the entire computational region. This is an attractive advantage of the finite volume method. There are some discrete methods, such as the finite difference method, where the discrete equations satisfy conservation of integrals only when the mesh is extremely fine; whereas the finite volume method shows accurate conservation of integrals even in the case of coarse meshes. As far as discretization methods are concerned, the finite volume method can be regarded as an intermediate between the finite unit method and the finite difference method. The finite cell method must assume the variation law of values between grid points (both interpolating functions) and use this as an approximate solution. The finite difference method considers only the values at the grid points and not how the values vary between grid points. The finite volume method seeks only the nodal values, which is similar to the finite difference method; however, the finite volume method must assume the distribution of values between grid points when seeking the integral of the control volume, which is again similar to the finite cell method. In the finite volume method, the interpolating function is only used to calculate the integral of the control volume, and after the discrete equation, the interpolating function can be forgotten; if necessary, different interpolating functions can be taken for different terms in the differential equation.