Gradient evolution-based efficient numerical calculation method for one-dimensional convective diffusion equation

Abstract

The invention belongs to the field of computational fluid mechanics, and discloses a gradient evolution-based one-dimensional convective diffusion equation efficient numerical calculation method, which comprises the following steps of: determining a steady solution and an analytical solution of a one-dimensional convective diffusion model equation set; setting a space gradient T of the variable u; establishing a gradient evolution method, and performing numerical calculation and post-processing; and establishing a gradient evolution equation. According to the gradient evolution-based efficient numerical calculation method for the one-dimensional convective diffusion equation, when the one-dimensional convective diffusion equation is solved, a correction function is manually added into a space gradient item, the original convective diffusion equation is converted into a convective equation set of an additional source item, and a second-order partial derivative diffusion item of the original equation is reduced into a first-order partial derivative; the stability condition is improved by one order of magnitude, the calculation burden is reduced, and the calculation efficiency is improved.

Description

Technical Field

[0001] The invention belongs to the field of computational fluid dynamics, and in particular relates to an efficient numerical calculation method for a one-dimensional convection-diffusion equation based on gradient evolution. Background Art

[0002] The numerical solution of partial differential equations has always been a hot issue in the field of computational fluid dynamics. Since the convection-diffusion equations (groups) have very wide applications in mechanics, environment, and engineering technology, the numerical solution of the convection-diffusion equations (groups) has certain practical significance, but in most cases, the analytical solution of the convection-diffusion equations (groups) is difficult to obtain. Based on the improvement of computer performance and the development of various numerical calculation theories and methods, seeking an effective, fast and stable numerical solution method for the convection-diffusion equations has become a hot topic at present.

[0003] There are three main methods for solving the convection-diffusion equations in engineering applications: the finite element method, the finite difference method, and the finite volume method. The basic idea of ​​these three mainstream methods is to mesh the computational domain, discretize the partial differential equations, transform the continuous equations into a finite set of linear algebraic equations, and find the approximate solution to the original problem, that is, to discretize the well-defined problem of the original partial differential equations, and then use the numerical approximation method to find the approximate solution to the original problem.

[0004] After the traditional solution method discretizes the convection-diffusion equation (group), its time asymptotic stability will be strictly limited by the grid density. Although many scholars have proposed many numerical calculation methods to solve numerical oscillation and numerical dispersion and improve calculation accuracy, such as characteristic line correction method, high-precision compact difference method, etc., various numerical calculation methods have increased the calculation complexity and reduced work efficiency.

[0005] Wind tunnel tests are an important means of studying the aerodynamic characteristics of aircraft in aerospace engineering. They can well simulate and measure the convection-diffusion phenomenon under actual flight conditions. In wind tunnel tests, simulating and measuring the convection-diffusion phenomenon in the flow field is crucial to understanding and predicting fluid dynamics behavior, and the one-dimensional convection-diffusion equation is the basic mathematical model to describe this physical phenomenon. In addition, the velocity field simulated in the wind tunnel test usually contains a variety of flow modes, such as laminar flow, turbulent flow, etc. The changes of these flow modes are complex and difficult to predict, and are often accompanied by a large amount of test data. However, wind tunnel tests are often accompanied by extremely high test costs. The use of efficient numerical calculation methods can greatly reduce the dependence on wind tunnel tests in practical applications. By inputting a small amount of data obtained in the wind tunnel test as the iterative initial field, the numerical model can quickly calculate the flow characteristics of the convection-diffusion flow field under steady convergence conditions, such as concentration distribution, temperature gradient, etc., which can not only reduce the test cost, but also greatly improve the calculation efficiency. At this time, an efficient and fast numerical solution method can significantly reduce the calculation time and improve the solution efficiency, allowing researchers to obtain test results faster and conduct in-depth analysis. In order to accurately extract information from the experimental data obtained from wind tunnel tests and predict the convection-diffusion behavior in the flow field, we need an effective, fast and stable numerical solution method.

[0006] Currently, there is an urgent need to develop an efficient numerical calculation method for the one-dimensional convection-diffusion equation based on gradient evolution. Summary of the invention

[0007] The technical problem to be solved by the present invention is to provide an efficient numerical calculation method for a one-dimensional convection-diffusion equation based on gradient evolution.

[0008] The high-efficiency numerical calculation method of the one-dimensional convection-diffusion equation based on gradient evolution of the present invention comprises the following steps: S10. Determine the steady-state and analytical solutions to the one-dimensional convection-diffusion model equations; The one-dimensional convection-diffusion model equations are as follows: ; The equation group (1) is a hyperbolic-parabolic equation, where It indicates the density or concentration of matter in space at a certain moment. represents a spatial variable, represents the time variable; is the convective diffusion coefficient, is the diffusion coefficient; , is a real number, is a positive real number; when hour, ; is the left boundary value, is the right boundary value; represents partial derivative; Assuming that the equation system (1) is a well-defined problem, the steady-state solution is: ; but: ; According to the theory of ordinary differential equations, the analytical solution is: ; in, is a natural constant; S20. Set the spatial gradient T of the variable u; ; Find: ; The following one-dimensional convection-diffusion model equations are derived: ; S30. Establishing a gradient evolution method and performing numerical calculation and post-processing; The time asymptotic method is used to solve the equation group (2). Since the second equation in the equation group (2) lacks a time term and the numerical stability of the one-dimensional convection-diffusion model equation group is limited by the Pe number, the function Adding a second equation improves the numerical stability of equation system (2); Then we obtain the one-dimensional convection-diffusion model equation system (3) with additional source terms: ; Perform first-order Euler discretization on equation system (3): ; in, , ; is the spatial grid point, is the number of time steps, is the time step, For the The spatial gradient at the spatial grid points, For the The spatial gradient at each spatial grid point; For the The spatial grid point values, For the The value of the spatial grid point; for and The spatial distance between For the Function of spatial grid points Numeric value; for Moment The spatial grid point values, for Moment The value of the spatial grid point; for Moment Function of spatial grid points Numeric value, for Moment Function of spatial grid points Numeric value; S40. Establish the gradient evolution equation; Conduct wind tunnel tests, establish a one-dimensional convection-diffusion flow field velocity field based on the measured data, draw the velocity-time curve of the one-dimensional convection-diffusion flow field velocity field, analyze the velocity distribution of the one-dimensional convection-diffusion flow field velocity field, determine that the flow field spatial gradient T and the variable u have a linear relationship, and introduce a linear function Establish the gradient evolution equation: ; but: ; For linear functions , under steady conditions: ,Right now ; The mathematical meaning is that the time change rate of the spatial derivative of variable u changes linearly with the time change rate of variable u, and the relationship between the time change rate of the spatial derivative of variable u and the time change rate of variable u under the steady-state convergence condition is established; The linear function Bring-in ,have: ; Substitute again , to obtain the linear function The gradient evolution equation is: .

[0009] The efficient numerical calculation method for one-dimensional convection-diffusion equation based on gradient evolution of the present invention transforms the original convection-diffusion equation into a group of convection equations with additional source terms by artificially adding a correction function in the spatial gradient term when solving the one-dimensional convection-diffusion equation, and reduces the second-order partial derivative diffusion term of the original equation to a first-order partial derivative, thereby improving the stability condition by an order of magnitude, reducing the computational burden, and improving the computational efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 This is a flow chart of the efficient numerical calculation method for the one-dimensional convection-diffusion equation based on gradient evolution of the present invention. DETAILED DESCRIPTION

[0011] The present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0012] Example: Figure 1 As shown, the efficient numerical calculation method of the one-dimensional convection-diffusion equation based on gradient evolution of this embodiment includes the following steps: S10. Determine the steady-state and analytical solutions to the one-dimensional convection-diffusion model equations; The one-dimensional convection-diffusion model equations are as follows: ; The equation group (1) is a hyperbolic-parabolic equation, where It indicates the density or concentration of matter in space at a certain moment. represents a spatial variable, represents the time variable; is the convective diffusion coefficient, is the diffusion coefficient; , is a real number, is a positive real number; when hour, ; is the left boundary value, is the right boundary value; represents partial derivative; Assuming that the equation system (1) is a well-defined problem, the steady-state solution is: ; but: ; According to the theory of ordinary differential equations, the analytical solution is: ; in, is a natural constant; S20. Set the spatial gradient T of the variable u; ; Find: ; The following one-dimensional convection-diffusion model equations are derived: ; S30. Establishing a gradient evolution method and performing numerical calculation and post-processing; The time asymptotic method is used to solve the equation group (2). Since the second equation in the equation group (2) lacks a time term and the numerical stability of the one-dimensional convection-diffusion model equation group is limited by the Pe number, the function Adding a second equation improves the numerical stability of equation system (2); Then we obtain the one-dimensional convection-diffusion model equation system (3) with additional source terms: ; Perform first-order Euler discretization on equation system (3): ; in, , ; is the spatial grid point, is the number of time steps, is the time step, For the The spatial gradient at the spatial grid points, For the The spatial gradient at each spatial grid point; For the The spatial grid point values, For the The value of the spatial grid point; for and The spatial distance between For the Function of spatial grid points Numeric value; for Moment The spatial grid point values, for Moment The value of the spatial grid point; for Moment Function of spatial grid points Numeric value, for Moment Function of spatial grid points Numeric value; S40. Establish the gradient evolution equation; Conduct wind tunnel tests, establish a one-dimensional convection-diffusion flow field velocity field based on the measured data, draw the velocity-time curve of the one-dimensional convection-diffusion flow field velocity field, analyze the velocity distribution of the one-dimensional convection-diffusion flow field velocity field, determine that the flow field spatial gradient T and the variable u have a linear relationship, and introduce a linear function Establish the gradient evolution equation: ; but: ; For linear functions , under steady conditions: ,Right now ; The mathematical meaning is that the time change rate of the spatial derivative of variable u changes linearly with the time change rate of variable u, and the relationship between the time change rate of the spatial derivative of variable u and the time change rate of variable u under the steady-state convergence condition is established; The linear function Bring-in ,have: ; Substitute again , to obtain the linear function The gradient evolution equation is: .

[0013] Although the embodiments of the present invention have been disclosed as above, they are not limited to the applications listed in the description and the embodiments. For those familiar with the art, all the features disclosed in the present invention, or the steps in all the methods or processes disclosed, except for mutually exclusive features and / or steps, can be combined in any way without departing from the principles of the present invention. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and the illustrations shown and described herein.

Claims

1. An efficient numerical calculation method for one-dimensional convection-diffusion equation based on gradient evolution, characterized in that: The following steps are involved: S10. Determine the steady-state and analytical solutions to the one-dimensional convection-diffusion model equations; The one-dimensional convection-diffusion model equations are as follows: ； The equation group (1) is a hyperbolic-parabolic equation, where It indicates the density or concentration of matter in space at a certain moment. represents a spatial variable, represents the time variable; is the convective diffusion coefficient, is the diffusion coefficient; , is a real number, is a positive real number; when hour, ; is the left boundary value, is the right boundary value; represents partial derivative; Assuming that the equation system (1) is a well-defined problem, the steady-state solution is: ； but: ； According to the theory of ordinary differential equations, the analytical solution is: ； in, is a natural constant; S20. Set the spatial gradient T of the variable u; ； Find: ； The following one-dimensional convection-diffusion model equations are derived: ； S30. Establishing a gradient evolution method and performing numerical calculation and post-processing; The time asymptotic method is used to solve the equation group (2). Since the second equation in the equation group (2) lacks a time term and the numerical stability of the one-dimensional convection-diffusion model equation group is limited by the Pe number, the function Adding a second equation improves the numerical stability of equation system (2); Then we obtain the one-dimensional convection-diffusion model equation system (3) with additional source terms: ； Perform first-order Euler discretization on equation system (3): ； in, , ; is the spatial grid point, is the number of time steps, is the time step, For the The spatial gradient at the spatial grid points, For the The spatial gradient at each spatial grid point; For the The spatial grid point values, For the The value of the spatial grid point; for and The spatial distance between For the Function of spatial grid points Numeric value; for Moment The spatial grid point values, for Moment The value of the spatial grid point; for Moment Function of spatial grid points Numeric value, for Moment Function of spatial grid points Numeric value; S40. Establish the gradient evolution equation; Conduct wind tunnel tests, establish a one-dimensional convection-diffusion flow field velocity field based on the measured data, draw the velocity-time curve of the one-dimensional convection-diffusion flow field velocity field, analyze the velocity distribution of the one-dimensional convection-diffusion flow field velocity field, determine that the flow field spatial gradient T and the variable u have a linear relationship, and introduce a linear function Establish the gradient evolution equation: ； but: ； For linear functions , under steady conditions: ,Right now ; The mathematical meaning is that the time rate of change of the spatial derivative of variable u changes linearly with the time rate of change of variable u, and the relationship between the time rate of change of the spatial derivative of variable u and the time rate of change of variable u under the steady-state convergence condition is established; The linear function Bring-in ,have: ； Substitute again , to obtain the linear function The gradient evolution equation is: 。

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