CN110069854A - Multiple resolution TWENO format is to the analogy method that can press flow field problems - Google Patents

Multiple resolution TWENO format is to the analogy method that can press flow field problems Download PDF

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CN110069854A
CN110069854A CN201910321659.XA CN201910321659A CN110069854A CN 110069854 A CN110069854 A CN 110069854A CN 201910321659 A CN201910321659 A CN 201910321659A CN 110069854 A CN110069854 A CN 110069854A
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王延萌
朱君
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Nanjing University of Aeronautics and Astronautics
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Abstract

Multiple resolution TWENO format is to the analogy method that can press flow field problems, and this method propose a kind of completely new finite difference multi-resolution trigonometric function weighted essential non-oscillatory schemes, can press flow field problems for numerical simulation.Its innovation is: only using the information on space center's Problem Representation, it is indicated without introducing any equivalent multi-resolution, use it is linear power no longer need that theoretic optimal solution is calculated by lengthy and tedious numerical value, can artificially be set as satisfaction and for 1 any positive number.The present invention is easier compared to classical WENO format, and robustness is stronger, and is easier to be generalized to higher dimensional space.Finally, the present invention sufficiently demonstrates the validity and reliability of this method using such novel finite difference multi-resolution TWENO format effectively numerical simulation Hyperbolic Conservation equation problem of multiple classics.

Description

Multiple resolution TWENO format is to the analogy method that can press flow field problems
Technical field
The invention belongs to Fluid Mechanics Computation field of engineering technology, and in particular to multiple resolution (multi-resolution) TWENO format is a kind of novel finite difference multi-resolution trigonometric function to the analogy method that can press flow field problems The calculation method of weighted essential non-oscillatory schemes under multinomial frame.
Background technique
In engineer application, construction solves the problems, such as the robustness of aerodynamic force, and the method for numerical simulation of high-efficiency high-accuracy is extremely It closes important.Nineteen fifty-nine, Godunov are that solution flow field problems propose the numerical simulation format of single order precision.The numerical value of single order precision Analogy method is not in non-physical numerical oscillation in capturing shock but can excessively smooth out strong discontinuity, and strong discontinuity is to problem Follow-up study important in inhibiting, therefore need imported high precision numerical value computational format simulate strong discontinuity class problem.In order to improve The precision of numeric format, simulates formal similarity and accurate capturing shock position, Harten have been put forward for the first time TVD in nineteen eighty-three (Total Variation Diminishing) format, and ENO was proposed in 1987 with Osher on this basis (Essentially Non-Oscillatory) format.The main thought of ENO format is in all centers and inclined space template It selects most smooth structure of transvers plate multinomial to find out the value at elementary boundary, and then reaches higher order accuracy in smooth domain, simultaneously The property of basic dead-beat is kept near strong discontinuity.Liu, Osher and Chan etc. proposed WENO (Weighted in 1994 Essentially Non-oscillatory) format, the utilization rate of space template is improved, and make the ENO of r rank precision Format can be increased to r+1 rank precision.1996, Jiang and Shu further improved WENO format, enabled numerical precision It is increased to 2r-1 rank, and designs the tectonic framework of new smooth indicator and nonlinear weight.The main thought of WENO format is logical The linear convex combination for crossing low order reconstruct flux obtains high-order approximation.Then, improve and have the advantages that different best WENO lattice Formula, declaration form tune WENO format mix tight WENO format, and the level of robust WENO format, limitation DG numerical solution reconstructs (HR) method With center WENO (CWENO) format.It is non-for the numerical simulation of different problems and different types of ENO and WENO format at present Often success.
It is known that multi-resolution format can reduce the calculating cost of high resolution scheme and high accurate scheme. Hyperbolic Conservation non trivial solution may include strong discontinuity in small and isolated region, may in remaining most of zoning It is smooth, therefore multi-resolution technology can concentrate on the region comprising strong discontinuity.Such multi- The original idea of resolution method is that Harten takes the lead in proposing to solve Hyperbolic Conservation equation.Give one by The function u (x) that the cell-average value in unit that unstructured grid is constituted indicates, Hartem are illustrated how the function point Solution is at different variation scales.By considering one group of nested grid, wherein given grid is best, and in each position It identifies grid most thick in set, can therefrom be restored to defined precision.Harten answers this multi-resolution For basic dead-beat (ENO) format, numerical flux calculation times needed for promoting a time step with reduction.This is to pass through Numerical solution when each time step is started is decomposed into resolution levels to realize, and it is appropriate compared in coarse grid each Position executes calculating.Dahmen is equal to the multi-resolution method for analyzing hyperbolic rule equation for 2001.2003, Chiavassa etc. has studied the multi-resolution method of the Hyperbolic Conservation equation based on adaptive format.2007, B ü rger etc. is to understand polymorphic type movement flow model to propose five rank WENO formats of multi-resolution technology.It is overall next It says, the purpose with multi-resolution technology is calculation amount to be concentrated mainly on the zonule comprising strong discontinuity.The present invention First design a new class of finite difference WENO format, smooth domain with one any center die plates and one 3 points of center Three ranks of the information acquisition object element boundary point in template are approximate, and when there is interruption in three dot center's templates, give up three The information of point template and approximate rank is reduced to 1 rank.If solving smooth enough, useable definition is in the two center die plates and 5 points Five ranks that information in heart template obtains elementary boundary point are approximate.Under identical condition, with these three space center's templates and Information in seven dot center's templates can get seven ranks approximation, and there is no this in intermittent situation in maximum space center die plates Method can be used for designing any high-order approximation.It does so, only with the point of the numerical solution of the central space template of Problem Representation Value does not introduce the multi-resolution statement of any equivalence.
Numerical method based on trigonometric polynomial function is suitable for simulating higher-order of oscillation problem and wave class phenomenon.Even if algebra Polynomial Reconstructing is a good building module for numerical flux, but it cannot be according to given higher-order of oscillation data Feature and be modified.When being inserted into the data of the higher-order of oscillation, the numeric format based on trigonometric polynomial reconstruct compares algebraic polynomial The numeric format of reconstruct is more suitable for solving this kind of higher-order of oscillation class problem.Let us recalls the hair of trigonometric polynomial reconstruct below Open up history.1976, Baron had studied trigonometric function interpolation and gives Neville class interpolation method.From 1973 to 1979 Year, Muhlbach also studied general basic function, including newton trigonometric function interpolation form.However, these sides above-mentioned Method cannot be directly used to ENO class interpolation scheme, because these methods are unsatisfactory for once increasing the rule of an interpolation point into template Then.In order to make up this defect, Christofi proposed a new class of local triangle polynomial interopolation method in 1966, can To increase interpolation point into template one by one and construct corresponding trigonometric function ENO format.2006, Yuan and Shu were mentioned Go out based on the DG method comprising index and the approximation space of trigonometric polynomial function.Later Zhu and Qiu was proposed in 2010 Five rank finite difference trigonometric function WENO format solution Hyperbolic Conservation equations and higher-order of oscillation class problem.2013, Zhu and Qiu It proposes a new class of limited bulk trigonometric function WENO format and he is regarded as the controller of Runge-Kutta DG method.Most Closely, Ha etc. proposes the improved ENO and WENO format based on exponential polynomials Function Solution Hyperbolic Conservation equation.It is general next It says, these ENO and WENO formats can improve the performance near smooth domain and interruption, than the numerical value side based on algebraic polynomial Method effect is more preferable.
Summary of the invention
The present invention aiming at the shortcomings in the prior art, provides a kind of multiple resolution TWENO format to can press flow field problems Analogy method provides in this method under a kind of completely new finite difference multi-resolution trigonometric function multinomial frame Weighted essential non-oscillatory schemes analog format, can for it is various press flow field problems especially higher-order of oscillation class problem carry out High resolution numerical simulation.On the one hand, with the weighted essential non-oscillatory schemes of trigonometric function polynomial construction ratio algebraic polynomial The problem of classics of construction add weighted essentially non-oscillatory (WENO) format to be easier to analog wave class or contain higher-order of oscillation class, and this is new Type finite difference multi-resolution TWENO format can obtain higher order values precision in smooth domain, in shock wave and connect Touching discontinuities can effectively inhibit the pseudo- appearance vibrated.On the other hand, novel finite difference multi-resolution TWENO Although the five identical space large form of rank finite difference WENO (WENO-JS) format of classics of format and Jiang and Shu, But the lower overall situation L1 and L ∞ norm truncated error can be obtained.
The present invention gives the novel finite difference multi-resolution that solution under cartesian grid can press flow field problems The construction process of the WENO highly-accurate nephelometric titrimetry format of trigonometric function spatially.It, should compared to classical WENO-JS format TWENO format by using the trigonometric function multinomial of reconstruct rather than reconstruct algebraic polynomial as finite difference WENO format Building module, solve the numerical simulation for pressing flow field problems of wave class and higher-order of oscillation class, and can reach in smooth domain To higher order values precision, format precision can constantly be reduced to single order numerical precision and keep basic dead-beat near strong discontinuity Characteristic.New TWENO format only uses the information on space center's Problem Representation, without introducing any equivalent multi- Resolution indicates that the linear power of use no longer needs that optimal solution fruit is calculated by lengthy and tedious numerical value, can be set as meeting With any positive number for 1.Novel finite difference multi-resolution TWENO format proposed by the present invention is compared to classics WENO-JS format is easier, and robustness is stronger, and is easier to be generalized to higher dimensional space.
To achieve the above object, the invention adopts the following technical scheme:
Multiple resolution TWENO format is to the analogy method that can press flow field problems, which comprises the steps of:
Step 1: the discrete finite difference scheme discrete for space half of Hyperbolic Conservation equation, using finite difference The approximation of multi-resolution TWENO format reconstruct flux；
Step 2: use the three discrete formula of rank TVD Runge-Kutta by half Discrete Finite difference of space in time orientation Format is separated into space-time approximate shceme finite difference scheme；
Step 3: the approximation on future time layer is obtained according to space-time approximate shceme finite difference scheme, successively iteration, obtains The numerical simulation in end time flow field in zoning.
To optimize above-mentioned technical proposal, the concrete measure taken further include:
Further, in the step 1, One-dimensional Hyperbolic Conservation Law Equations are as follows:
The form of its space semi-discrete scheme are as follows:
Wherein, utU is indicated to t derivation, u indicates that variable function, t indicate time variable, fx(u) indicate f (u) to x derivation, f (u) flux function, x representation space variable, u are indicated0Indicate initial state value, u (x, 0) and u0(x) initial time variable letter is indicated The expression formula of number u, L (u) expression-fx(u) spatial spreading form；
Spatial spreading at the grid cell of uniform lengthElement lengthIn unit The heart, that is, least bit isWherein i is coordinate serial number, is had:
Wherein,WithFlux f (u) is respectively indicated in grid cell IiBoundaryWithThe numerical flux at place, with EnsureIn point x=xiSame order precision approximation is sentenced to fx(u), ui(t) point value u (x is indicatedi, t) accurately solve Approximation；
Ask flux f (u) in object element IiBoundaryWithThe high-order approximation value at placeWith
To ensure correctly tendency and stability windward, flux f (u) is split into f (u)=f+(u)+f-(u), whereinThen with the approximate each single item of wind direction difference of own；It is divided using Lax-FriedrichsWhereinTherefore, numerical flux divides are as follows:
Then using the approximation of finite difference multi-resolution TWENO format reconstruct flux, with three ranks, five The higher order accuracy finite difference multi-resolution TWENO format of rank and seven ranks is constructed.
Further, the higher order accuracy finite difference multi-resolution with three ranks, five ranks and seven ranks The construction process of TWENO format, specific as follows:
Step 1: selecting a series of central space templates, and the trigonometric function of reconstruct different accuracy is more in different templates Item formula；
Step 2: obtaining the polynomial equivalent expression of trigonometric function of the different accuracy of reconstruct；
Step 3: taking the linear power of satisfaction and any positive number for 1 first, then calculate smooth indicator, for measuring three Angle function reconstructs smoothness of the multinomial on object element；
Step 4: calculating nonlinear weight on the basis of linear power and smooth indicator；
Step 5: finding out numerical flux in the approximation of object element boundary, and then obtain half Discrete Finite difference of space Format.
Further, the step 1 is specific as follows:
Step 1.1: for three rank spaces approximation, selecting two spaces center die plates T1=[Ii,] and T2=[Ii-1, Ii, Ii+1]；Then multinomial q is reconstructed on trigonometric function space1(x) ∈ span { 1 } and trigonometric function multinomialSo that:
Step 1.2: for five rank spaces approximation, selecting space center's template T3=[Ii-2..., Ii+2]；Then in triangle Trigonometric function multinomial is reconstructed on function space:
So that:
Step 1.3: for seven rank spaces approximation, selecting space center's template T4=[Ii-3..., Ii+3]；Then in triangle Trigonometric function multinomial is reconstructed on function space:
So that:
Further, in the step 2, p is enabled1(x)=q1(x) and meet:
WhereinIndicate linear power,And
The polynomial equivalent expression of different trigonometric functions of reconstruct is obtained, detailed process is as follows:
Step 2.1: for three ranks approximation, trigonometric function multinomial p2(x) it indicates are as follows:
Wherein γ1,2+γ2,2=1 and γ2,2≠0；
Step 2.2: for five ranks approximation, trigonometric function multinomial p3(x) it indicates are as follows:
WhereinAnd γ3,3≠0；
Step 2.3: for seven ranks approximation, trigonometric function multinomial p4(x) it indicates are as follows:
WhereinAnd γ4,4≠0。
Further, in the step 3, if linear power isWhereinAnd l2=2,3,4；
Calculate smooth indicatorFor measuring trigonometric function multinomialRaw section On smoothness, calculation formula are as follows:
Wherein κ=2 (l2-1)；
For smooth indicator β1, it is zoomed into a value from zero according to following definition, is defined first:
Wherein m=1,2,3, respectively correspond that three ranks, five ranks and seven rank precision are approximate, on ε indicates the positive number of very little to prevent The denominator of formula is zero；Then it sets:
Further, it in the step 4, using WENO-Z method, defines:
Nonlinear weight are as follows:
Further, in the step 5, numerical flux division f is found out+(u) pointMulti-resolution The trigonometric function of TWENO format reconstructs multinomial:
Numerical fluxReconstruct aboutIt is mirror-symmetrical.
Further, in the step 2, the calculated result of numerical flux is substituted into the space containing time-derivative item half Discrete Finite difference scheme obtains the ODE about time-derivative；
Utilize the three discrete formula of rank TVD Runge-Kutta:
Obtain space-time approximate shceme finite difference scheme, wherein u(1), u(2)For intermediate form, Δ t is time step, on Marking n indicates the n-th time horizon, L (un), L (u(1)), L (u(2)) it is-fx(u) approximation of high order spatial discrete form.
Further, in the step 3, space-time approximate shceme finite difference scheme is the iterative formula about time horizon, just Beginning state value successively obtains in end time zoning it is known that find out the approximation of future time layer by iterative formula Numerical simulation.
The beneficial effects of the present invention are: compared to WEN0 format, the TWENO format pass through trigonometric function multinomial without It is building module of the algebraic polynomial as finite difference TWENO format, simulating wave class and higher-order of oscillation class can press flow field to ask Topic, while higher order accuracy can be reached in smooth domain；Compared to existing trigonometric function Polynomial Reconstructing format, the TWENO lattice The global L that formula is obtained using nested central space template and multi-resolution technology is layered1Truncated error and L∞Truncation Error is smaller, while also avoiding generating Nonphysical Oscillation at intense shock wave and contact discontinuity, the relation line in the TWENO format Property power no longer need by it is complicated be calculated but be set as and for 1 any positive number, therefore new TWENO format construction It is simpler, it is easier to be applied in the numerical simulation of higher dimensional space.
Detailed description of the invention
Fig. 1 a to 1f indicates that the One-Dimensional Euler equation containing mobile sine wave, numerical value calculate density, T=2.Fig. 1 a, 1b Using third-order format, Fig. 1 c, 1d use five rank precision formats, and Fig. 1 e, 1f use seven rank precision formats；Fig. 1 a, 1c, 1e In, k=3 takes 70 grids；In Fig. 1 b, 1d, 1f, k=5 takes 130 grids；In each figure, solid line indicates accurate solution, and plus sige is indicated with new The density that type TWENO format obtains, square indicate the density obtained with classics WENO-JS format.
Fig. 2 a to 2f indicates shock wave and vortex interferes with each other problem, and numerical value calculates pressure, 30 pressure contours, value model It encloses for 1.02-1.33, T=0.35.Fig. 2 a, 2b use third-order format, and Fig. 2 c, 2d use five rank precision formats, Fig. 2 e, 2f Using seven rank precision formats, 120 × 60 grid；Fig. 2 a, 2c, 2e are the results obtained with novel TWENO format；Fig. 2 b, 2d, 2f is the result obtained with classics WENO-JS format.
Fig. 3 a to 3f indicates shock wave and vortex interferes with each other problem, and numerical value calculates pressure, 90 pressure contours, value model It encloses for 1.19-1.37, T=0.6.Fig. 3 a, 3b use third-order format, and Fig. 3 c, 3d use five rank precision formats, Fig. 3 e, 3f Using seven rank precision formats, 120 × 60 grid；Fig. 3 a, 3c, 3e are the results obtained with novel TWENO format；Fig. 3 b, 3d, 3f is the result obtained with classics WENO-JS format.
Fig. 4 a to 4f indicates shock wave and vortex interferes with each other problem, and numerical value calculates pressure, 30 pressure contours, value model It encloses for 1.1-1.3, T=0.8.Fig. 4 a, 4b use third-order format, and Fig. 4 c, 4d use five rank precision formats, and Fig. 4 e, 4f are adopted With seven rank precision formats, 120 × 60 grid；Fig. 4 a, 4c, 4e are the results obtained with novel TWENO format；Fig. 4 b, 4d, 4f It is the result obtained with classics WENO-JS format.
Specific embodiment
In conjunction with the accompanying drawings, the present invention is further explained in detail.
One-dimensional Hyperbolic Conservation Law Equations are considered first:
The form of its space semi-discrete scheme are as follows:
Wherein, utU is indicated to t derivation, u indicates that variable function, t indicate time variable, fx(u) indicate f (u) to x derivation, f (u) flux function, x representation space variable, u are indicated0Indicate initial state value, u (x, 0) and u0(x) initial time variable letter is indicated The expression formula of number u, L (u) expression-fx(u) spatial spreading form.
For the sake of simplicity, spatial spreading at the grid cell of uniform lengthElement lengthUnit center, that is, least bit isWherein i is coordinate serial number, ui(t) point value u (x is indicatedi, T) the approximation accurately solved, has:
Wherein,WithFlux f (u) is respectively indicated in grid cell IiBoundaryWithThree ranks, five ranks at place Or seven rank precision numerical flux, to ensureIn point x=xiSame order precision approximation is sentenced to fx(u)；ui(t) table Show u in grid cell IiInterior point xiValue u (the x at placei, t).
Ask flux f (u) in object element IiBoundaryWithThe high-order approximation value at placeWithSpecific steps are such as Under:
To ensure correctly tendency and stability windward, flux f (u) is split into f (u)=f+(u)+f-(u), whereinThen with the approximate each single item of wind direction difference of own.The simple Lax- of this patent Friedrichs divisionWhereinTherefore, numerical flux can divide are as follows:
Below with novel finite difference multi-resolution TWENO format reconstruct flux approximation, this patent with For the construction process of the higher order accuracy finite difference multi-resolution TWENO format of three ranks, five ranks and seven ranks, it is For the sake of simplicity, this patent only describes f+(u) pointThe restructuring procedure at place is simultaneously defined as
Step 1: selecting a series of central space templates, and reconstruct the trigonometric function multinomial of different numbers, detailed process It is as follows:
Step 1.1: for three rank spaces approximation, we select two spaces center die plates T1=[Ii,] and T2=[Ii-1, Ii, Ii+1].Then multinomial q is reconstructed on trigonometric function space1(x) ∈ span { 1 } and trigonometric function multinomialSo that:
And
Step 1.2: for five rank spaces approximation, we select space center template T3=[Ii-2..., Ii+2].Then exist Trigonometric function spatially reconstructs trigonometric function multinomial
So that:
Step 1.3: for seven rank spaces approximation, we select space center template T4=[Ii-3..., Ii+3].Then exist Trigonometric function spatially reconstructs trigonometric function multinomial
So that:
Step 2: obtaining the polynomial equivalent expression of different trigonometric functions of reconstruct.For unified symbol, p is enabled1(x) =q1(x) and
WhereinAndDetailed process is as follows:
Step 2.1: for three ranks approximation, trigonometric function multinomial p2(x) it may be expressed as:
Wherein γ1,2+γ2,2=1, and γ2,2≠0。
Step 2.2: for five ranks approximation, trigonometric function multinomial p3(x) it may be expressed as:
WhereinAnd γ3,3≠0。
Step 2.3: for seven ranks approximation, trigonometric function multinomial p4(x) it may be expressed as:
WhereinAnd γ4,4≠0。
Step 3: taking the linear power of satisfaction and any positive number for 1 first, avoid complicated numerical procedure, then count Smooth indicator is calculated, for measuring smoothness of the trigonometric function reconstruct multinomial on object element.
In previous stepWherein l=1 ..., l2And l2=2,3,4, it is linearly to weigh.Based on Non-smooth surface region intense shock wave The accuracy of transmission and smooth domain with basic dead-beat wave, according to existing research this patent set linear power asWhereinAnd l2=2,3,4.For example, approximate for third-order true WithHave for five rank precision approximations WithHave for seven rank precision approximationsWith
Calculate smooth indicatorFor measuring trigonometric function multinomialIn section On smoothness, calculation formula are as follows:
Wherein κ=2 (l2-1).For smooth indicator β1, it can be zoomed into a value from zero according to following definition. It defines first
Wherein m=1,2,3 (it is approximate to respectively correspond three ranks, five ranks and seven rank precision), ε indicates the positive number of very little to prevent (17) denominator of formula is zero.Then it sets
Step 4: calculating nonlinear weight on the basis of linear power and smooth indicator.
Using WENO-Z method, definition
Nonlinear weight are as follows:
ε=10 are taken in the simulation process of this patent-4。
Step 5: finding out numerical flux division f+(u) pointMulti-resolution TWENO format three Angle function reconstructs multinomial:
Numerical fluxReconstruct aboutIt is mirror-symmetrical.
Secondly, by calculated result substitute into the half Discrete Finite difference scheme of space containing time-derivative item, obtain about when Between derivative ODE.
Finally, utilizing the three discrete formula of rank TVD Runge-Kutta:
Obtain space-time approximate shceme finite difference scheme, wherein u(1), u(2)For intermediate form, Δ t is time step, on Marking n indicates the n-th time horizon, L (un), L (u(1)), L (u(2)) it is-fx(u) approximation of high order spatial discrete form.
Space-time approximate shceme finite difference scheme is the iterative formula about time horizon, and initial state value is it is known that pass through iteration Formula finds out the approximation of future time layer, successively obtains the numerical simulation in end time zoning.
For two-dimensional problems, by the restructuring procedure above Wesy.
From the foregoing, it will be observed that the major advantage of this method is: being reconstructed with the novel multi-resolution TWENO format Information when multinomial on the Problem Representation of use space center does not introduce any equivalent multi-resolution table Show；Used in format it is linear power no longer need that optimal solution is calculated by lengthy and tedious numerical value, can be set as meet and for 1 appoint Meaning positive number；The problem of suitable for wave class or containing higher-order of oscillation class；Format can obtain higher order values precision in smooth domain, The progressive depression of order of numerical precision energy is single order numerical precision and the characteristic that can keep basic dead-beat at shock wave and contact discontinuity.
Specific implementation example of several examples as this patent disclosed method is given below.
Example 1: solution one-dimensional nonlinear Burgers ' equation:
Its primary condition is u (x, 0)=0.5+sin (π x), meets periodic boundary condition.When numerical value calculating t=0.5/ π Solution.The error and numerical precision such as table 1 obtained with novel finite difference multi-resolution TWENO format numerical simulation It is shown.Compare for convenience, the error and numerical precision obtained with classical WENO-JS format numerical simulation is also found in table 1.
Table 1
Example 2: solution two-dimension non linearity Burgers ' equation:
Its primary condition is u (x, y, 0)=0.5+sin (π (x+y)/2), meets periodic boundary condition.Numerical value calculates t= Solution when 0.5/ π.The error and numerical value obtained with novel finite difference multi-resolution TWENO format numerical simulation Precision is as shown in table 2.Compare for convenience, the error and numerical precision obtained with classical WENO-JS format numerical simulation also arranges In table 2.
Table 2
Example 3: solution One-Dimensional Euler equation:
Wherein density when ρ, u are the speed in the direction x, and E is gross energy, and p is pressure.Primary condition is ρ (x, 0)=1+ =1, γ=1.4 0.2sin (x), u (x, 0)=1, p (x, 0).The zoning of x is [0,2 π], meets periodic boundary condition.It is close The accurate solution of degree is ρ (x, t)=1+0.2sin (x-t), and numerical value calculates solution when t=2.With novel finite difference multi- The error and numerical precision that resolution TWENO format numerical simulation obtains are as shown in table 3.Compare for convenience, with classics The error and numerical precision that WENO-JS format numerical simulation obtains are also found in table 3.
Table 3
Example 4: solution Two-dimensional Euler Equations:
Wherein ρ is density, and u is the speed in the direction x, and v is the speed in the direction y, and E is gross energy, and p is pressure.Primary condition It is ρ (x, y, 0)=1+0.2sin (x+y), u (x, y, 0)=1, v (x, y, 0)=1, p (x, y, 0)=1, γ=1.4.Calculate area Domain is (x, y) ∈ [0,2 π] × [0,2 π], all meets periodic boundary condition in two directions.The accurate solution of density is ρ (x, y, t)=1+0.2sin (x+y-2t), numerical value calculate solution when t=2.With novel finite difference multi-resolution The error and numerical precision that TWENO format numerical simulation obtains are as shown in table 4.Compare for convenience, with classical WENO-JS format The error and numerical precision that numerical simulation obtains are also found in table 4.
Table 4
Example 5: the One-Dimensional Euler equation containing mobile sine wave in solution density:
Numerical value calculates solution when t=6.With novel finite difference multi-resolution TWENO format and with classics As shown in Figure 1, wherein k=3,5, grid number takes respectively for density that the density and accurate solution of WENO-JS format numerical simulation obtain 70 and 130.As shown in Fig. 1 a to 1f, in this example, novel finite difference multi-resolution TWENO format is than classical WENO-JS formatted analog effect is more preferable, especially at wave crest and trough.
6. shock wave of example and vortex interfere with each other problem.The shock wave of Mach 2 ship 1.1 is located at x=0.5 and perpendicular to x-axis. Shock wave original state isSmall vortex is located at the left side of the shock wave and its center is located at (xc, yc) At=(0.25,0.5).Vortex can regard the speed of mean flow as, and the disturbance of temperature and entropy indicates are as follows:
Wherein τ=r/rc,ε=0.3, rc=0.05, α=0.204, γ=1.4.Meter Calculation region is [0,2] × [0,1].In the pressure isogram in [0,1] × [0,1] region when Fig. 2 a to 2f gives t=0.35. In the pressure isogram in [0.4,1.45] × [0,1] region when Fig. 3 a to 3f gives t=0.6.Fig. 4 a to 4f gives t= In the pressure isogram in [0,2] × [0,1] region when 0.8.As shown in Figures 2 to 4, in this example, novel finite difference Multi-resolution TWEN0 format is more preferable than classical WEN0-JS formatted analog effect, especially at vortex and shock wave.
It should be noted that the term of such as "upper", "lower", "left", "right", "front", "rear" cited in invention, also Only being illustrated convenient for narration, rather than to limit the scope of the invention, relativeness is altered or modified, in nothing Under essence change technology contents, when being also considered as the enforceable scope of the present invention.
The above is only the preferred embodiment of the present invention, protection scope of the present invention is not limited merely to above-described embodiment, All technical solutions belonged under thinking of the present invention all belong to the scope of protection of the present invention.It should be pointed out that for the art For those of ordinary skill, several improvements and modifications without departing from the principles of the present invention should be regarded as protection of the invention Range.

Claims (10)

1. multiple resolution TWENO format is to the analogy method that can press flow field problems, which comprises the steps of:
Step 1: the discrete finite difference scheme discrete for space half of Hyperbolic Conservation equation, using finite difference multi- The approximation of resolution TWENO format reconstruct flux；
Step 2: use the three discrete formula of rank TVD Runge-Kutta by half Discrete Finite difference scheme of space in time orientation It is separated into space-time approximate shceme finite difference scheme；
Step 3: the approximation on future time layer is obtained according to space-time approximate shceme finite difference scheme, successively iteration, is counted Calculate the numerical simulation in end time flow field in region.
2. multiple resolution TWENO format as described in claim 1 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 1, One-dimensional Hyperbolic Conservation Law Equations are as follows:
The form of its space semi-discrete scheme are as follows:
Wherein, utU is indicated to t derivation, u indicates that variable function, t indicate time variable, fx(u) indicate f (u) to x derivation, f (u) Indicate flux function, x representation space variable, u0Indicate initial state value, u (x, 0) and u0(x) initial time variable function u is indicated Expression formula, L (u) expression-fx(u) spatial spreading form；
Spatial spreading at the grid cell of uniform lengthElement lengthUnit center i.e. half It puts and isWherein i is coordinate serial number, is had:
Wherein,WithFlux f (u) is respectively indicated in grid cell IiBoundaryWithThe numerical flux at place, to ensureIn point x=xiSame order precision approximation is sentenced to fx(u), ui(t) point value u (x is indicatedi, t) the approximation accurately solved Value；
Ask flux f (u) in object element IiBoundaryWithThe high-order approximation value at placeWith
To ensure correctly tendency and stability windward, flux f (u) is split into f (u)=f+(u)+f-(u), whereinThen with the approximate each single item of wind direction difference of own；Using Lax-Friedrichs points It splitsWhereinTherefore, numerical flux divides are as follows:
Then using finite difference multi-resolution TWENO format reconstruct flux approximation, with three ranks, five ranks and The higher order accuracy finite difference multi-resolution TWENO format of seven ranks is constructed.
3. multiple resolution TWENO format as claimed in claim 2 is to the analogy method that can press flow field problems, it is characterised in that:
The construction of the higher order accuracy finite difference multi-resolution TWENO format with three ranks, five ranks and seven ranks Journey, specific as follows:
Step 1: selecting a series of central space templates, and the trigonometric function of reconstruct different accuracy is multinomial in different templates Formula；
Step 2: obtaining the polynomial equivalent expression of trigonometric function of the different accuracy of reconstruct；
Step 3: taking the linear power of satisfaction and any positive number for 1 first, smooth indicator is then calculated, for measuring triangle letter Smoothness of the number reconstruct multinomial on object element；
Step 4: calculating nonlinear weight on the basis of linear power and smooth indicator；
Step 5: finding out numerical flux in the approximation of object element boundary, and then obtain half Discrete Finite difference lattice of space Formula.
4. multiple resolution TWENO format as claimed in claim 3 is to the analogy method that can press flow field problems, it is characterised in that:
The step 1 is specific as follows:
Step 1.1: for three rank spaces approximation, selecting two spaces center die plates T1=[Ii,] and T2=[Ii-1, Ii, Ii+1]；So Multinomial q is reconstructed on trigonometric function space afterwards1(x) ∈ span { 1 } and trigonometric function multinomialSo that:
Step 1.2: for five rank spaces approximation, selecting space center's template T3=[Ii-2..., Ii+2]；Then in trigonometric function Spatially reconstruct trigonometric function multinomial:
So that:
Step 1.3: for seven rank spaces approximation, selecting space center's template T4=[Ii-3..., Ii+3]；Then in trigonometric function Spatially reconstruct trigonometric function multinomial:
So that:
5. multiple resolution TWENO format as claimed in claim 4 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 2, p is enabled1(x)=q1(x) and meet:
WhereinIndicate linear power,And
The polynomial equivalent expression of different trigonometric functions of reconstruct is obtained, detailed process is as follows:
Step 2.1: for three ranks approximation, trigonometric function multinomial p2(x) it indicates are as follows:
Wherein γ1,2+γ2,2=1 and γ2,2≠0；
Step 2.2: for five ranks approximation, trigonometric function multinomial p3(x) it indicates are as follows:
WhereinAnd γ3,3≠0；
Step 2.3: for seven ranks approximation, trigonometric function multinomial p4(x) it indicates are as follows:
WhereinAnd γ4,4≠0。
6. multiple resolution TWENO format as claimed in claim 5 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 3, if linear power isWhereinAnd l2=2,3,4；
Calculate smooth indicatorFor measuring trigonometric function multinomialIn sectionOn Smoothness, calculation formula are as follows:
Wherein κ=2 (l2—1)；
For smooth indicator β1, it is zoomed into a value from zero according to following definition, is defined first:
Wherein m=1,2,3, respectively corresponding three ranks, five ranks and seven rank precision approximations, ε indicates the positive number of very little to prevent above formula Denominator is zero；Then it sets:
7. multiple resolution TWENO format as claimed in claim 6 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 4, using WENO-Z method, define:
Nonlinear weight are as follows:
8. multiple resolution TWENO format as claimed in claim 7 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 5, numerical flux division f is found out+(u) pointMulti-resolution TWENO format Trigonometric function reconstructs multinomial:
Numerical fluxReconstruct aboutIt is mirror-symmetrical.
9. multiple resolution TWENO format as claimed in claim 8 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 2, the calculated result of numerical flux is substituted into the half Discrete Finite difference lattice of space containing time-derivative item Formula obtains the ODE about time-derivative；
Utilize the three discrete formula of rank TVD Runge-Kutta:
Obtain space-time approximate shceme finite difference scheme, wherein u(1), u(2)For intermediate form, Δ t is time step, subscript n table Show the n-th time horizon, L (un), L (u(1)), L (u(2)) it is-fx(u) approximation of high order spatial discrete form.
10. multiple resolution TWENO format as claimed in claim 9 is to the analogy method that can press flow field problems, it is characterised in that:
In the step 3, space-time approximate shceme finite difference scheme is iterative formula about time horizon, initial state value it is known that The approximation of future time layer is found out by iterative formula, successively obtains the numerical simulation in end time zoning.
CN201910321659.XA 2019-04-22 2019-04-22 Multiple resolution TWENO format is to the analogy method that can press flow field problems Pending CN110069854A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110457806A (en) * 2019-08-02 2019-11-15 南京航空航天大学 The whole flow field analogy method of five rank WENO format of center based on staggered-mesh
CN110781626A (en) * 2019-10-31 2020-02-11 南京航空航天大学 Simulation method of finite difference multiple resolution trigonometric function WENO format
CN111177965A (en) * 2019-12-25 2020-05-19 南京航空航天大学 Multi-resolution WENO format fixed-point fast scanning method based on solving of constancy problem
CN111177965B (en) * 2019-12-25 2022-06-17 南京航空航天大学 Multi-resolution WENO format fixed-point fast scanning method based on solving of constancy problem
CN112163312A (en) * 2020-08-17 2021-01-01 空气动力学国家重点实验室 Method for carrying out numerical simulation on compressible flow problem through high-order WENO format reduction
CN112307684A (en) * 2020-10-16 2021-02-02 南京航空航天大学 Fixed-point fast scanning method combining multi-resolution WENO format with ILW boundary processing

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