Method for simplifying conventional numerical integration in discrete velocity space

Abstract

The invention discloses a method for simplifying traditional numerical integration in a discrete speed space, which is based on MCI integration, selects discrete speed coordinate points by means of the golden section number theory integration principle, introduces the consistent distribution theory of the number theory into a multi-integration calculation method, and simplifies the traditional numerical integration. According to the method for simplifying the traditional numerical integration in the discrete speed space, the traditional numerical integration for the discrete speed space is simplified, the calculation efficiency of multiple integrals in high-dimensional flow is improved, and particularly, the distribution of discrete speed coordinate points is optimized; and an effective technical approach is provided for developing a novel Boltzmann equation collision integral calculation method.

Description

technical field

[0001] The invention belongs to the technical field of aircraft aerodynamics, and mainly relates to a numerical integration method, which is used to simplify traditional numerical integration in discrete velocity spaces. Background technique

[0002] Computational fluid dynamics is a subject that solves the governing equations of fluid mechanics by numerical methods, obtains a discrete quantitative description of the flow field, and uses this to predict the law of fluid motion. In CFD, the integral and differential terms in the governing equations of fluid motion are approximately expressed as discrete algebraic forms (roughly meaning converting continuous curves into points), so that the integral or differential governing equations are transformed into algebraic equations Then, solve these algebraic equations by computer, so as to obtain the numerical solution of the flow field at discrete time / space points. CFD is also sometimes called numerical simulation...

Examples

Embodiment

[0085] According to the error estimation principle of the number theory method, for any integer n>3, use the golden section number theory integration method to perform the approximate calculation of the double integral of s=2, as follows:

[0086]

[0087] In the formula, a(>1) and C(>0) are constants, and E(a, C) is the integrand f(x, y), which constitutes a function class. The analysis shows that the error limit of the above integration method is

[0088] But for the trapezoidal formula with s=2, let n=m 2 , then there are:

[0089]

[0090] This shows that the error term given by the traditional numerical integration method than the error term given by the number-theoretic integration method Much worse.

[0091] Specifically, we take representative values ​​of n and α respectively (1≤n≤10, α takes values ​​1, 1.5, 2, 2.5, 3) as Figure 1-Figure 5 The comparison chart of the present invention explains the advantages of the present invention. By observing the a...

technical field

[0001] The invention belongs to the technical field of aircraft aerodynamics, and mainly relates to a numerical integration method, which is used to simplify traditional numerical integration in discrete velocity spaces. Background technique

[0002] Computational fluid dynamics is a subject that solves the governing equations of fluid mechanics by numerical methods, obtains a discrete quantitative description of the flow field, and uses this to predict the law of fluid motion. In CFD, the integral and differential terms in the governing equations of fluid motion are approximately expressed as discrete algebraic forms (roughly meaning converting continuous curves into points), so that the integral or differential governing equations are transformed into algebraic equations Then, solve these algebraic equations by computer, so as to obtain the numerical solution of the flow field at discrete time / space points. CFD is also sometimes called numerical simulation...

Examples

Embodiment

[0085] According to the error estimation principle of the number theory method, for any integer n>3, use the golden section number theory integration method to perform the approximate calculation of the double integral of s=2, as follows:

[0086]

[0087] In the formula, a(>1) and C(>0) are constants, and E(a, C) is the integrand f(x, y), which constitutes a function class. The analysis shows that the error limit of the above integration method is

[0088] But for the trapezoidal formula with s=2, let n=m 2 , then there are:

[0089]

[0090] This shows that the error term given by the traditional numerical integration method than the error term given by the number-theoretic integration method Much worse.

[0091] Specifically, we take representative values ​​of n and α respectively (1≤n≤10, α takes values ​​1, 1.5, 2, 2.5, 3) as Figure 1-Figure 5 The comparison chart of the present invention explains the advantages of the present invention. By observing the a...

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