Boundary type numerical method for effectively improving calculation efficiency of multi-dimensional convection diffusion problem

Abstract

The invention belongs to the field of numerical method calculation efficiency optimization, and particularly relates to a boundary type calculation method of a coupling alternating algorithm. Taking a two-dimensional convection diffusion equation as an example, the main idea of the numerical calculation method is as follows: 1, carrying out local one-dimensional discretization on a solved two-dimensional problem; 2, reducing the order of the differential equation by using a gradient variable; 3, deriving the relationship between the concentration and the concentration gradient at two adjacent nodes through integration, and further obtaining the variable relationship between any node and a boundary node; 3, solving a result in one direction according to the idea of a one-dimensional problem; and solving the result in the next direction by taking the result as a new initial condition. The method can greatly improve the calculation efficiency under the condition of ensuring the calculation precision, has the characteristic of small occupied computer memory, is suitable for three-dimensional problems, is favorable for processing numerical simulation problems of large-scale grids, and has important significance for practical engineering application.

Description

Technical Field

[0001] The invention belongs to the field of optimizing the computational efficiency of numerical methods, and in particular relates to a boundary type computational method coupled with an alternating algorithm. Background Art

[0002] In the research process, experimental methods and numerical simulation methods are two important research methods. Compared with experimental methods, numerical simulation methods have the advantages of saving money, short experimental cycle, adjustable parameters, and multiple uses; in addition, the results obtained by numerical simulation methods are more convenient to view, and changing calculation parameters will not greatly increase resource consumption. Therefore, today's numerical simulation methods have entered a stage of vigorous development, and many scientific researchers are still working hard to study more accurate, faster, and more suitable numerical simulation methods for different engineering situations. It is of great significance to develop efficient and accurate numerical calculation methods.

[0003] The half-boundary method (HBM) has been applied in the fields of heat and mass transfer and nuclear physics due to its advantages such as easy handling of boundary conditions and small storage unit, such as nuclear fuel melting phase change, convection diffusion of pollutants, and multi-group neutron diffusion. Summary of the invention

[0004] Purpose of the invention: To develop a boundary type calculation method coupled with an alternating algorithm, the main purpose of which is to greatly improve the calculation efficiency.

[0005] Technical solution: In order to achieve the above-mentioned purpose, the present invention adopts the following technical solution.

[0006] A coupled calculation method for simulating the solution of the convection-diffusion equation is characterized by combining the local one-dimensional method with the semi-boundary method to transform a two-dimensional problem into two one-dimensional problems; at the same time, when applying the semi-boundary method to solve the one-dimensional equation, the gradient variable is introduced to reduce the order of the differential equation, and the second-order convection-diffusion equation is reduced to a first-order equation system, and the relationship between the node at the boundary and any node is derived, thereby reducing the matrix order in the calculation process, reducing the calculation storage, and improving the calculation efficiency. Taking the two-dimensional problem as an example, the method is implemented in the following steps in the calculation process: Step 1: perform local one-dimensional discretization on the two-dimensional problem to be solved, as shown below: in, is the unknown quantity, u is the velocity, and Γ is the diffusion coefficient. For the result at time (k+1), we can assume that the time integration domain of the first equation is [k, k+1 / 2] and the time integration domain of the second equation is [k+1 / 2, k+1]. Step 2: Use the newly introduced gradient variable to reduce the order of the split one-dimensional equation, and transform a second-order partial differential equation with one dependent variable and two independent variables into a first-order partial differential equation system with two dependent variables and two independent variables, as shown below: Among them, U and V are introduced gradient variables. Step 3: Integrate the equations with respect to time and space respectively to obtain the relationship between variables in time and space. The relationship between the obtained adjacent node variables is organized into a matrix form, which can be expressed as in, β t =Δt / 2, Step 4, take a set of matrix relationships as an example and organize them into: It is not difficult to see that through recursion, the relationship between the variables at any node and the boundary node at different times can be obtained, thereby quickly realizing the calculation.

[0007] The beneficial effects of the present invention are: (1) Improve computational efficiency by reducing a two-dimensional problem to two one-dimensional problems, and confirm the computational advantages of the coupled algorithm by comparing it with other computational methods; (2) In the process of solving one-dimensional problems, the order of the largest matrix involved does not increase with the increase of the number of grids, and is always second order, so it takes up less computer memory and takes less computing time; (3) Applicable to the case of discontinuous parameters; (4) Applicable to three-dimensional problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 The basic steps for solving convection-diffusion problems using the new boundary-type method; Figure 2 The temperature distribution solved by the boundary algorithm at t = 10.0s; Figure 3 The results of the boundary-type algorithm and the finite volume method are compared under different grids. DETAILED DESCRIPTION

[0009] The present invention provides an efficient and accurate boundary type numerical algorithm, which is further described below in conjunction with the accompanying drawings and specific examples.

[0010] Examples Consider a point in time around the origin The transport process of the distributed heat source under the action of convection and diffusion is determined by the following well-defined problem: The exact solution to the problem is Among them, the diffusion coefficient Γ = 0.06W / (m·K), the velocity u = v = 0.006m / s, L = H = 30m, and the first-class boundary conditions used can be obtained from the exact solution. In order to compare the computational efficiency of the coupled algorithm, the solution is based on the finite volume method at the same time. The convection term uses the second-order upwind difference format, and the diffusion term uses the central difference format. The time step is set to 0.1s and the grid size is 1m.

[0011] Solution: The coupling algorithm is applied to solve the internal temperature distribution and compared with the exact solution to verify the accuracy of the boundary-type algorithm; by comparing with the results of the finite volume method, the storage and efficiency advantages of the algorithm are analyzed. Among them, the error is the solution of the algorithm, and φ(i,j) is the exact solution. It can be seen that the algorithm has good accuracy and computational efficiency. At the same time, as the number of grids increases, the advantage of computational efficiency becomes more obvious.

Claims

1. After the boundary conditions and initial conditions are known, a coupling algorithm based on a boundary type method and a local one-dimensional method is used to quickly and accurately solve the problem, which is characterized by The two-dimensional problem is split into two one-dimensional problems, and then the gradient variables are used to reduce the order of the two one-dimensional differential equations respectively, and the relationship between the variables at two adjacent nodes in a single direction is derived by integration, and then the relationship between the variables at any node and the boundary node is obtained. After solving the variables in the two directions in turn, the purpose is finally achieved. The method is specifically implemented in the following steps during the calculation process: Step 1, apply the local one-dimensional method to transform a two-dimensional problem into two one-dimensional problems; Step 2, using the newly introduced gradient variables to reduce the order of the two one-dimensional convection-diffusion equations respectively, and transforming a second-order partial differential equation with one variable and two independent variables into a first-order partial differential equation system with two variables and two independent variables; Step 3: Integrate the equations in the two directions respectively, and integrate the two equations in the equations with respect to time and space respectively to obtain the relationship between the two variables in time and space: Determine the time step Δt and space step Δx, Δy, where t is the number of time steps, and use superscripts k, k+1, k+2...k+t to represent the time dimension at each node; Step 4: For the first direction, the variable value at the initial time (0 time) is considered as the initial value. The relationship between the concentration and the concentration gradient at the adjacent nodes at time k+t / 2 can be obtained through integration operation. Step 5: For the second direction, the variable value at the initial time (k+t / 2) is considered as the initial value. The relationship between the concentration and the concentration gradient at the adjacent nodes at the k+t time can be obtained by integration operation. Step 6: Finally, by repeatedly repeating steps 4 and 5, the variables at the final moment are solved.

2. A coupled numerical method capable of solving the convection-diffusion equation according to claim 1, characterized in that: By splitting the two-dimensional problem into a one-dimensional problem, and then introducing gradient variables and constructing a relationship matrix, the computational efficiency can be greatly improved while ensuring the computational accuracy.

3. The novel calculation method according to claim 1, characterized in that: It is also applicable to three-dimensional problems and is beneficial for engineering applications.

4. The novel calculation method according to claim 1, characterized in that: When solving one-dimensional equations, the matrix used by the computer to run the calculations involved in this method is always second-order and does not increase with the number of grids, which can reduce storage requirements.

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