CN107133418A - Fluids material stratospheric transport analogy method based on alternately TVD difference algorithms - Google Patents
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Abstract
The invention discloses the advection transport equation that a kind of Fluids material stratospheric transport analogy method based on alternately TVD difference algorithms includes building Fluids material, and then numerical discretization is carried out to the advection transport equation, obtain the discrete equation of Fluids material conveying；The discrete equation that Fluids material is conveyed disassembles into the step number equal with its dimension；Obtain the flow field data of Fluids material and input and iterative numerical calculating is carried out in the discrete equation conveyed to Fluids material, predict the Fluids material concentration in each default iteration time section；During iterative numerical is calculated, different types of total variance abatement restricted function is used alternatingly according to the step number equal with its dimension in each dimension respectively in different default iteration time sections to calculate Fluids material concentration in next period, the change of Fluids material stratospheric transport process medium fluid material concentration is finally simulated and predict.
Description
Technical field
Calculated the invention belongs to fluid dynamic numerical simulation and calculating field, more particularly to a kind of alternately TVD difference that is based on
The Fluids material stratospheric transport analogy method of method.
Background technology
Numerical model is research and the important work for forecasting the motion such as river, ocean and its interior transport of substances in nature
Tool, with the development of computer technology and social economy, numerical model is in scientific research, operational forecast and Hu Ku, ocean etc.
The effect become more and more important is played in the industry such as environmental management and planning.
Advection is that the main process that material carries out space conveying is carried in Fluids, and advection process is represented by equation
(1).It is generally using the method for difference that the progress of (1) formula is discrete in numerical model, obtain such as the advection difference side of formula (2) form
Journey, then carries out iterative numerical to formula (2), obtains material concentration c spatial distribution and change over time.
Wherein, c transports material concentration for certain, and u, v, w are respectively the flow velocity in x, y, z direction.
Wherein,For the material concentration at half storey grid, Δ t is discrete
Time step, Δ x, Δ y, Δ z are respectively the grid spacing in x, y, z direction, and label subscript n represents time step number, subscript i,
J, k represent the grid label in x, y, z direction respectively.
In difference equation (2), the material concentration at half storey grid how is calculatedIt is the main difference of different difference schemes
Where not.For example, upstreame scheme uses upstream grid concentration, middle poor form uses being averaged for upstream and downstream grid
Concentration.Although these algorithms calculate easy, the problem of there is low precision or strong numerical solidification, cause analog result distortion compared with
It is strong and the stability of model can be influenceed.
Total variance abatement (Total Variation Diminishing, abbreviation TVD) difference scheme is the high-precision nothing of a class
The advection difference algorithm of frequency dispersion, it is widely used in ocean and Atmospheric Numerical Model and correlative study.TVD algorithms are mainly
The high-precision item of back-diffusion is added on the basis of original low precision upstreame scheme, the precision of advection form is improved, draws simultaneously
Enter restricted function to control the intensity of high-precision item added, with this, keep in whole integral process forward, numeric format it is total
It is deteriorated and keeps not increasing, so as to effectively suppresses the numerical solidification that high accurate scheme generally has.
The algorithm of TVD advection forms is following (by taking x directions as an example)：
As shown in formula (3), calculatingWhen, plus high-precision correction term on the basis of upstreame scheme, and before correction term
Add restricted function ψ (ri+1/2), thus obtain：
Wherein,
Add after restricted function and ensure the total variance of result of calculationHolding does not increase, i.e.,：
TV(cn+1)≤TV(cn) (4)
Restricted function need to be caused to meet following conditions：
Wherein, r represents concentration gradient ratio, i.e.,Or
Although TVD restricted functions have a variety of, a kind of restricted function of single use can not be controlled in TVD forms well
The intensity of the high-precision item of addition, thereby resulted in above TVD numeric formats still have certain numerical d ispersion (e.g., Minmod,
Van Leer, MUSCL, HSIMT) and the pseudo- effect of back-diffusion (e.g., Superbee).In addition, TVD forms can ensure one-dimensional before
In the case of total variance do not increase, it is and then barely satisfactory in the case of multidimensional, however it remains numerical solidification.Therefore, existing TVD is utilized
When advection difference scheme to the transport of substances in Fluids simulate for a long time, unsatisfactory simulation knot is still suffered from
Really.
The content of the invention
In order to solve the deficiencies in the prior art, the invention provides a kind of Fluids based on alternately TVD difference algorithms
Material stratospheric transport analogy method, it provides a kind of effectively decrease numerical d ispersion and the pseudo- effect of back-diffusion and solves multidimensional frequency dispersion and puts down
Difference method-the TVDal (TVD with Alternating Limiters) of stream format, it is right in Fluids to be applied to
The simulation and forecast of transport of substances.
A kind of Fluids material stratospheric transport analogy method based on alternately TVD difference algorithms of the present invention, including：
The advection transport equation of Fluids material is built, and then numerical discretization is carried out to the advection transport equation, is obtained
The discrete equation conveyed to Fluids material；The dimension of the discrete equation of the Fluids material conveying is at least two dimension；
The discrete equation that Fluids material is conveyed disassembles into the step number equal with its dimension；
Obtain the flow field data of Fluids material and input in the discrete equation conveyed to Fluids material and enter line number
Value iterative calculation, predicts the Fluids material concentration in each default iteration time section；
During iterative numerical is calculated, in different default iteration time sections respectively in each dimension according to
The equal step number of its dimension is used alternatingly different types of total variance abatement restricted function to calculate the earth in next period
Flowing material concentration, finally simulates and predicts the change of Fluids material stratospheric transport process medium fluid material concentration.
Further, the building method based on the abatement form that is deteriorated, the advection transport equation to earth flowing material is carried out
Numerical discretization.
Further, total variance abatement restricted function includes the restricted function and numerical value back-diffusion property of numerical d ispersion property
Restricted function.
Further, Fluids material concentration is the function on time and space.
Further, the process for asking for the advection transport equation of Fluids material is：
Fluids material concentration is respectively to the derived function for the directioin parameter for characterizing space, then the side with corresponding sign space
After being multiplied to the flow velocity of parameter, the Fluids material concentration that finally adds up obtains Fluids material to the derived function of time
Advection transport equation.
Wherein, the restricted function of the numerical d ispersion property is Minmod functions.
The restricted function of the numerical d ispersion property is Van Leer functions.
The restricted function of the numerical d ispersion property is MUSCL functions.
The restricted function of the numerical d ispersion property is HSIMT functions.
The restricted function of the numerical value back-diffusion property includes Superbee functions.
Wherein, actually calculate in, frequently with several classical TVD restricted functions mainly have：
Superbee：
ψ(r)Superbee=max [0, min (2r, 1), min (r, 2)] (6)
Van Leer(or Harmonic)：
ψ(r)VanLeer=(r+ | r |)/(r+1) (7)
MUSCL：
Minmod：
ψ(r)Minmod=max [0, min (r, 1)] (9)
HSIMT：
ψ(r)HSIMT=max [0, min (2r, 2, β)] (10)
Wherein,
Compared with prior art, the beneficial effects of the invention are as follows：
(1) this method does not change original formula increase extra computation amount, but is used alternatingly not in different time step
Connatural algorithm, largely reduces the numerical d ispersion and the pseudo- effect of back-diffusion of traditional TVD advections difference algorithm so that
The error of analog result is smaller.
(2) alternately restricted function is split with discrete equation in this method and be combined, it is flat without frequency dispersion based on obtained multidimensional
Flow algorithm, effectively eliminates the TVD advection forms numerical solidification problem that transport of substances is simulated in the case of Multi-dimension calculation.
Brief description of the drawings
The Figure of description for constituting the part of the application is used for providing further understanding of the present application, and the application's shows
Meaning property embodiment and its illustrate be used for explain the application, do not constitute the improper restriction to the application.
Fig. 1 is the schematic flow sheet of the present invention；
Fig. 2 (a) is initial time material concentration distribution map；
Fig. 2 (b) is the Superbee analog results based on traditional TVD algorithms；
Fig. 2 (c) is the HSIMT analog results based on traditional TVD algorithms；
Fig. 2 (d) is the Superbee analog results that traditional TVD algorithms combination discrete equation splits algorithm；
Fig. 2 (e) is the HSIMT analog results that traditional TVD algorithms combination discrete equation splits algorithm；
Fig. 2 (f) is the analog result of the inventive method.
Embodiment
It is noted that described further below is all exemplary, it is intended to provide further instruction to the application.Unless another
Indicate, all technologies used herein and scientific terminology are with usual with the application person of an ordinary skill in the technical field
The identical meanings of understanding.
It should be noted that term used herein above is merely to describe embodiment, and be not intended to restricted root
According to the illustrative embodiments of the application.As used herein, unless the context clearly indicates otherwise, otherwise singulative
It is also intended to include plural form, additionally, it should be understood that, when in this manual using term "comprising" and/or " bag
Include " when, it indicates existing characteristics, step, operation, device, component and/or combinations thereof.
The dimension of discrete equation of Fluids material conveying is at least two dimension, below with Fluids material convey from
Exemplified by dissipating the dimension of equation and being two dimension：
As shown in figure 1, the Fluids material stratospheric transport analogy method based on alternately TVD difference algorithms of the present invention,
Including：
Step 1：Build Fluids material advection transport equation, and then to the advection transport equation carry out numerical value from
Dissipate, obtain the discrete equation of Fluids material conveying.
The advection transport equation for building Fluids material is formula (1)：
Numerical discretization is carried out to the advection transport equation, the discrete equation for obtaining the conveying of Fluids material is formula
(2)：
Wherein,For the material concentration at half storey grid, Δ t is discrete time step,
Δ x, Δ y are respectively the grid spacing in x, y direction, and label subscript n represents time step number, and subscript i, j represents x, y direction respectively
Grid label.
Step 2：The discrete equation that Fluids material is conveyed disassembles into the step number equal with its dimension.
Step 3：Obtain the flow field data of Fluids material and input in the discrete equation conveyed to Fluids material
Iterative numerical calculating is carried out, the Fluids material concentration in each default iteration time section is predicted；
During iterative numerical is calculated, in different default iteration time sections respectively in each dimension according to
The equal step number of its dimension is used alternatingly different types of total variance abatement restricted function to calculate the earth in next period
Flowing material concentration, finally simulates and predicts the change of Fluids material stratospheric transport process medium fluid material concentration.
Specifically, flow velocity u, v information is inputted, the parity of iterative steps is iterated and judged to discrete equation, if
Odd number is calculated then to be calculated using below step a), if even number is then calculated using step b).
Step a)：
Discrete equation formula (2) is disassembled as formula (3-1) and the calculating of the step of formula (3-2) two：
Wherein,Subscript com and dif represent respectively using numerical value back-diffusion property (for example, Superbee) and numerical value expansion
The TVD restricted functions of property (for example, Minmod) are dissipated to calculate material concentration at half storey grid
Discrete equation is disassembled into (3-1) and (3-2) two step, advection process in x and y both directions is calculated successively：First
When step calculates advection process on x directions, first using a kind of TVD restricted functions of numerical value back-diffusion property, calculating obtains middle change
Measure c*；Intermediate variable c* is substituted into y directions difference equation by second step, is calculated advection process on y directions, is used a kind of numerical d ispersion
The TVD restricted functions of property calculate the concentration c for obtaining new time stepn+1。
Step b)：
Equally it is divided to (4-1) and (4-2) two step to calculate, but the first step first calculates y directions advection process, it is anti-using a kind of numerical value
The TVD restricted functions of scattering nature, calculating obtains intermediate variable c*；Intermediate variable c* is substituted into x directions difference equation by second step,
X equation advection processes are calculated, using a kind of TVD restricted functions of numerical d ispersion property, the material that calculating obtains new time step is dense
Degree.
Wherein,Subscript com and dif represent respectively using numerical value back-diffusion property (for example, Superbee) and numerical value expansion
The TVD restricted functions of property (for example, Minmod) are dissipated to calculate material concentration at half storey grid
Finally judge to integrate the setting value whether duration reaches pattern, no then return to step 2 continues, terminated if reaching
Calculate.
Present disclosure can be further illustrated by following analog result.
1st, simulated experiment：Using the inventive method and other method, the material advection process to giving a certain concentration distribution
Simulated.
2nd, analog result
Exist in a horizontal two-dimension marine site, in it with 12 hours for reciprocal flowing of the cycle along 45 ° of directions, initial time
Material concentration distribution is given to be distributed as shown in Fig. 2 (a), and knot is simulated after 100 cycles obtained using different advection difference algorithms
Fruit is respectively as shown in Fig. 2 (b)-Fig. 2 (f).Wherein, Fig. 2 (b) and 2 (c) are based on traditional TVD algorithms Superbee and HSIMT
Analog result, Fig. 2 (d) and Superbee and the HSIMT simulation that Fig. 2 (e) is that tradition TVD algorithm combinations discrete equation splits algorithm
As a result；Fig. 2 (f) is the analog result of the inventive method.
From Fig. 2 (a)-Fig. 2 (f) as can be seen that the analog result based on traditional TVD algorithms has obvious numerical value frequency
Dissipate, splitting numerical solidification after algorithm using discrete equation is improved, but still suffers from the pseudo- effect of numerical value back-diffusion and diffusion, is made
Then maintain the distribution of true solution substantially with the inventive method, reduce the pseudo- effect of numerical value back-diffusion and diffusion of traditional TVD algorithms,
Result of calculation achieves gratifying analog result without numerical solidification simultaneously.
To sum up, this method proposed by the present invention effectively weakens numerical d ispersion and the pseudo- effect of back-diffusion and solves multidimensional frequency dispersion
Difference algorithm-the TVDal of advection form material advection analogy method.The method of the present invention is conversely pseudo- by the way that property is used alternatingly
Traditional TVD restricted functions of effect, effectively reduce numerical d ispersion and back-diffusion, reduce the error of advection analog result,
Meanwhile, Multidimensional Discrete equation is subjected to fractionation calculating, it is ensured that multidimensional advective simulation is that no numerical solidification is produced, and the above is improved logical
The analog result for crossing the experiment of material advection can be seen that.
Although above-mentioned the embodiment of the present invention is described with reference to accompanying drawing, not to present invention protection model
The limitation enclosed, one of ordinary skill in the art should be understood that on the basis of technical scheme those skilled in the art are not
Need to pay various modifications or deform still within protection scope of the present invention that creative work can make.
Claims (10)
1. a kind of Fluids material stratospheric transport analogy method based on alternately TVD difference algorithms, it is characterised in that including：
The advection transport equation of Fluids material is built, and then numerical discretization is carried out to the advection transport equation, obtains ground
The discrete equation of ball flowing material conveying；The dimension of the discrete equation of the Fluids material conveying is at least two dimension；
The discrete equation that Fluids material is conveyed disassembles into the step number equal with its dimension；
Obtain the flow field data of Fluids material and input progress numerical value in the discrete equation conveyed to Fluids material and change
In generation, calculates, and predicts the Fluids material concentration in each default iteration time section；
During iterative numerical is calculated, tieed up respectively in each dimension according to it in different default iteration time sections
The equal step number of number is used alternatingly different types of total variance abatement restricted function to calculate Fluids in next period
Material concentration, finally simulates and predicts the change of Fluids material stratospheric transport process medium fluid material concentration.
2. the Fluids material stratospheric transport analogy method as claimed in claim 1 based on alternately TVD difference algorithms, it is special
Levy and be, based on the building method for the abatement form that is deteriorated, the advection transport equation to earth flowing material carries out numerical discretization.
3. the Fluids material stratospheric transport analogy method as claimed in claim 1 based on alternately TVD difference algorithms, it is special
Levy and be, total variance abatement restricted function includes the restricted function of numerical d ispersion property and the limitation letter of numerical value back-diffusion property
Number.
4. the Fluids material stratospheric transport analogy method as claimed in claim 1 based on alternately TVD difference algorithms, it is special
Levy and be, Fluids material concentration is the function on time and space.
5. the Fluids material stratospheric transport analogy method as claimed in claim 4 based on alternately TVD difference algorithms, it is special
Levy and be, the process for asking for the advection transport equation of Fluids material is：
Fluids material concentration is joined respectively to the derived function for the directioin parameter for characterizing space, then with the corresponding direction for characterizing space
After several flow velocitys is multiplied, the Fluids material concentration that finally adds up obtains the advection of Fluids material to the derived function of time
Transport equation.
6. the Fluids material stratospheric transport analogy method as claimed in claim 3 based on alternately TVD difference algorithms, it is special
Levy and be, the restricted function of the numerical d ispersion property is for Minmod functions or from his scattering nature restricted function.
7. the Fluids material stratospheric transport analogy method as claimed in claim 3 based on alternately TVD difference algorithms, it is special
Levy and be, the restricted function of the numerical d ispersion property is for Van Leer functions or from his scattering nature restricted function.
8. the Fluids material stratospheric transport analogy method as claimed in claim 3 based on alternately TVD difference algorithms, it is special
Levy and be, the restricted function of the numerical d ispersion property is for MUSCL functions or from his scattering nature restricted function.
9. the Fluids material stratospheric transport analogy method as claimed in claim 3 based on alternately TVD difference algorithms, it is special
Levy and be, the restricted function of the numerical d ispersion property is for HSIMT functions or from his scattering nature restricted function.
10. the Fluids material stratospheric transport analogy method as claimed in claim 3 based on alternately TVD difference algorithms, its
It is characterised by, the restricted function of the numerical value back-diffusion property includes Superbee functions or limits letter from his back-diffusion property
Number.
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| CN114840974A (en) * | 2022-03-25 | 2022-08-02 | 中国气象局地球***数值预报中心 | Advection mode system suitable for complex terrain and operation method thereof |
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