CN108229083A - A kind of Flow Numerical Simulation method based on improved finite difference scheme - Google Patents

A kind of Flow Numerical Simulation method based on improved finite difference scheme Download PDF

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CN108229083A
CN108229083A CN201810324786.0A CN201810324786A CN108229083A CN 108229083 A CN108229083 A CN 108229083A CN 201810324786 A CN201810324786 A CN 201810324786A CN 108229083 A CN108229083 A CN 108229083A
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王镇明
朱君
赵宁
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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Abstract

A kind of Flow Numerical Simulation method based on improved finite difference scheme, the five rank finite difference WENO forms taken can be appointed by relating to a kind of linear power, in numerical simulation shallow water problems, it need to only appoint and three positive numbers for summing to 1 is taken as linear power subsequently calculate.The numeric format that the present invention constructs can not only keep the five rank precision in the smooth waters of numerical simulation under arbitrary linear power, but also can be in the transition for being interrupted basic dead-beat at waters, so as to more accurately solve the speed in the depth and all directions of flow.In addition, it is of the invention herein in connection with well balanced technologies, so as to reduce the influence of riverbed topography logarithm form so that analog result is more accurate.Finally, the present invention verifies its superiority in shallow water problems calculating by the numerical example of various bed configurations.

Description

A kind of Flow Numerical Simulation method based on improved finite difference scheme
Technical field
The invention discloses a kind of Flow Numerical Simulation method based on improved finite difference scheme, available for calculating number The technical fields such as the hydraulic engineerings such as, tide simulation stream, dam break, flood.
Background technology
The water conservancies works such as shallow water equation (the also known as Saint-Venant equations) Yu Haiyang that is widely used, river, lake Journey field, it is also possible to carry out the natural disasters such as numerical simulation dam break, flood.Recent two decades, the numerical solution of shallow water equation It obtains extensive concern and is developed well.The main difficulty of numerical solution shallow water equation is：First, it belongs to non-thread Property Hyperbolic Conservation equation one kind, solution includes the labyrinths such as shock wave, strong discontinuity, so the requirement of logarithm form is very It is high；Second is that the source item on the right of equation is very complicated, and it is related with the topography in riverbed, so numeric format needs to meet under hydrostatic state Certain conservativeness.
Many scholars study this, and are successfully applied to relevant practical problem.1993, Alcrudo and Limited bulk Godunov forms are successfully applied to solve Wave of Two-Dimension Shallow Water Equation by Carcia-Navarro.Bermudez and Vazquez constructs a kind of upstreame scheme and solves shallow water equation, and defines numeric format for the first time shallow water equation must be expired " C-property " property of foot.In addition, the wave propagation scheme of also Leveque constructions, Capilla and Balaguer-Beser Roe forms of the limited bulk cell centered scheme of design, Castro et al. etc..
It is above-mentioned to be substantially single order or second order form for solve shallow water equation, and recent decades are directed to Hyperbolic conservation laws The higher order values of rule equation solve form and are always the research contents of emphasis and have obtained good development.1987, Harten ENO (the basic dead-beats of Essentially Non-Oscillatory) forms have been put forward for the first time with Osher and are successfully applied to count The related fields such as fluid operator mechanics.The main thought of ENO forms is that most smooth reconstruct target is selected in all candidate templates The value of unit, so as to fulfill high-precision and the property of basic dead-beat.And because of it there are stencil-chosen process so that meter The waste of result of calculation can be caused during calculation, so having constructed WENO (Weighted ENO) form on this basis. 1994, WENO forms were suggested and construct three rank limited bulk forms of one-dimensional situation for the first time.1996, Jiang and Shu 3 ranks and 5 rank finite difference WENO forms are constructed, and give the general Computational frame of WENO forms, including reconstructing multinomial Construction, the calculating linearly weighed, the calculating of smooth indicator and the calculating of nonlinear weight and final reconstruction value.Hereafter many Person is put into the theoretical research and engineer application of ENO and WENO forms.But until 2002, Vokovic and Sopta just will be upper ENO and WENO forms are stated to be applied to solve one-dimensional shallow water wave equation.And continue at 2004, will have with Crnjaric-Zic cooperations It limits volume WENO and center WENO forms is applied to solve shallow water equation and Open-Channel flow equations.Xing and Shu The finite difference WENO forms of satisfaction " C-property " property then are constructed, and passes through the numerical example and demonstrates it effectively Property.
Optimum linearity power in finite difference WENO form construction process uniquely determines, and otherwise can lose numeric format Optimal design design accuracy, specific value calculate it is relative complex, especially in Uneven mesh spacing.Therefore, Zhu and Qiu in The finite difference taken WENO forms can be appointed for solving the Euler in Fluid Mechanics Computation by devising within 2016 a kind of linear power Equation, and can ensure that numeric format can keep the optimal precision of smooth domain under arbitrary linear power.Based on this thought, The present invention is applied to solution shallow water equation, and combines the " C- that " well-balanced " technology ensures numeric format Property " properties finally meet a certain region of Runge-Kutta time discrete Method for Numerical Simulation of TVD properties using 3 ranks Interior streamflow regime, and by numerical experiments such as cylinder dam breaks come the feasibility of verification method.
Invention content
The present invention provides a kind of flow Numerical-Mode based on improved finite difference scheme for deficiency of the prior art Plan method.
To achieve the above object, the present invention uses following technical scheme：
A kind of Flow Numerical Simulation method based on improved finite difference scheme, which is characterized in that establish space right-angle Coordinate system, with uniform grid subdivision zoning, in the calculating grid of formation, the basic weighting dead-beat of 5 rank precision of construction Form carries out the Digital calculation modelling in zoning, specifically includes following steps：
Step 1: in the numerical computations region of waters, generate rectangular coordinate system and carry out mesh generation；
Step 2: the discrete shallow-water wave of finite difference WENO forms with 5 rank precision is constructed on the calculating grid of formation Space derivation item in equation group, the linear power in construction process can arbitrary value；
Step 3: the variation of circulation and riverbed gradient is balanced using well-balanced methods；
Step 4: shallow water equation group is carried out on time orientation using classical Runge-Kutta time discretes method Propulsion, so as to obtain the flow depth of any time, the speed of all directions and water flow, draw up reality so as to Numerical-Mode The streamflow regime at each moment in problem.
To optimize above-mentioned technical proposal, the concrete measure taken further includes：
In the step 1, carry out the respective grid points coordinate after mesh generation and be denoted as (xi, yj), wherein x, y representation spaces Variable, subscript i, j represent grid serial number.
In the step 2, the shallow water equation group for describing water movement is：
Above-mentioned equation group is uniformly write as：Ut+F(U)x+G(U)y=S, wherein, t represents time variable, x, y representation spaces Variable, U=(h, hu, hv)TRepresent conservation variable vector, F (U), G (U) represent flux vector, F (U)xRepresent F (U) to x derivations, G (U)yRepresent G (U) to y derivations, S=(0 ,-ghbx,- ghby)TRepresent riverbed effect item, h, u, v, b are the functions about time variable t and space variable x, y, represent flow respectively Depth, horizontal direction speed, vertical direction speed and bed level, bxRepresent b to x derivations, byB is represented to y derivations, T is represented Transposition, g represent acceleration of gravity.
For the shallow water equation group, linear power can appoint the construction process of finite difference WENO forms taken as follows：
1st step, to the space derivation item F (U) in equation groupxWith G (U)yCarrying out difference approximation has：
Wherein, (xi, yj) represent serial number i, j point coordinate,X directions and y directions are represented respectively Point (xi±1/2, yj) and (xi, yj±1/2) at numerical flux；
2nd step, to each coordinate points (xi, yj) at flux carry out Lax-Friedrichs thinking positions：
F(UI, j)=F (Ui, j)++F(UI, j)-,
G(UI, j)=G (UI, j)++G(UI, j)-,
Wherein, UI, jRepresent U in point (xi, yj) at functional value, αlRepresent the maximum eigenvalue of x directions flux matrix, αk Represent the maximum eigenvalue of y directions flux matrix；
3rd step passes through F (UI, j)±The linear power of value construction can appoint the new finite difference WENO forms taken, obtain F (UI+1/2, j)±The high-order approximation value at place similarly, passes through G (UI, j)±Be worth to G (UI+1/2, j)±Value.
3rd step specifically comprises the following steps：
Step 3.1 first fixes y=yj, take the template T of 5 unit spots0={ II-2, j, II-1, j, II, j,II+1, j, II+2, j, structure Make a quartic polynomialIt is another to take 2 template T for including 2 unit spots1={ II-1, j, II, jAnd T2={ II, j, II+1, j, construct two linear polynomialsWithMeet following constraints respectively：
By F±(UI, j) be denoted asThenExpression be：
Step 3.2 appoints the positive number γ for taking one group and being 10, γ1, γ2As linear power；
Step 3.3, the multinomial gone out according to three structure of transvers plateCalculate smooth indicator
Wherein, n represents corresponding polynomial serial number, and α represents summing target, and r represents corresponding polynomial number,Representative polynomialα partial derivative is asked to independent variable x；
ThenExpression be：
Step 3.4, linearly power and smooth indicator according to arbitrary value, calculate nonlinear weightIt calculates Formula is as follows：
Wherein,It represents to calculate transition value；
The nonlinear weight that step 3.5, the linear term formula obtained according to step 3.1 and step 3.4 obtain, obtains final F (UI+1/2, j)±High-order approximation value：
Step 3.6 fixes x=x againi, obtain G (UI, j+1/2)±High-order approximation value.
In the step 3, it is as follows using well-balanced methods：
1st step, the riverbed effect item S divided in shallow water equation group are as follows：
Wherein, S1, S2Represent two source items after division；
2nd step solves two source item S after division with finite difference WENO1, S2；
3rd step obtains half Discrete Finite difference scheme of flow cavitation result equation：
Wherein L (UI, j)=- F (U)x-G(U)y+S1+S2, represent the height obtained after being solved by finite difference WENO forms Rank approximation.
In the step 4, time step is Δ t, tnRepresent the n-th time horizon,Represent U (xi, yj, tn) value, in tn Time horizon is calculatedThen using the Runge-Kutta time discrete methods for meeting TVD properties discrete half from Finite difference scheme is dissipated so as to obtain space-time approximate shceme finite difference scheme, it is as follows：
Wherein,WithTo calculate intermediate transition value, direction in space and time orientation it is discrete after the completion of obtain Space-time approximate shceme finite difference scheme is the iterative formula about time horizon, it is known that depth of water h, the x directions of initial streamflow regime The water velocity v in water velocity u, y directions, then constantly recycles according to iterative formula, you can obtain flow at a time or The high-precision numerical approximation value of physical quantity during stable state.
The beneficial effects of the invention are as follows：Constructed in uniform coordinate grid linear power can arbitrary value finite difference WENO forms, and " well-balanced " technology is combined for solving with smooth or interruption riverbed shallow water flow problem.With biography The finite difference WENO numerical methods of system are compared, and improved five ranks finite difference WENO forms have the following advantages that：1) linear power Can arbitrary value, only need to meet sum to 1 positive number, and smooth domain be attained by optimal design essence Degree；2) linear power is unrelated with grid, under arbitrary grid can arbitrarily value without falling precision；3) convergence is more preferable, for fixed Chang Wenti can rapidly converge to accuracy of machines.In addition, the present invention has also combined " well-balanced " technology, so as to Preferably handle smooth or interruption riverbed shallow water flow problem.
Description of the drawings
Fig. 1 is the sectional view in riverbed and river.
Fig. 2 a to Fig. 2 e are the isopleth of water depth figures of smooth riverbed water surface perturbed problem different moments.
Fig. 3 is the schematic three dimensional views for being interrupted riverbed.
Fig. 4 a-4b are the isopleth of water depth figures for being interrupted riverbed water surface perturbed problem different moments.
Fig. 5 a-5f are the schematic three dimensional views of flat riverbed cylinder Dam Break Problems different moments.
Fig. 6 is the schematic three dimensional views of hump riverbed cylinder Dam Break Problems different moments.
Specific embodiment
In conjunction with the accompanying drawings, the present invention is further explained in detail.
Consider two-dimensional shallow water flow control equation, i.e. shallow water equation：
For convenience of statement, it is abbreviated as：
Ut+F(U)x+G(U)y=S (2)
Wherein, t represents time variable, x, y representation space variables, U=(h, hu, hv)TRepresent conservation variable vector,F (U), G (U) represent flux vector, F (U)xRepresent F (U) to x derivations, G (U)yRepresent G (U) to y derivations, S=(0 ,-ghbx,-ghby)TRepresent riverbed effect item, bxTable Show b to x derivations, byB is represented to y derivations, h, u, v, b represent respectively flow depth, horizontal direction speed, vertical direction speed with And bed level, and be all the function about time variable t and space variable x, y, T represents transposition, and g represents acceleration of gravity, Generally it is taken as 9.812m/s2。
It is worth noting that, other than only related with space variable in addition to b (x, y), above-mentioned all physical quantitys be all about The function of time variable and space variable changes with the variation of spatial position at any time.
Enable xi+1/2-xi1/2=Δ x, yj+1/2-yj-1/2=Δ y, grid element center areNet Lattice unit is II, j=[xi-1/2, xi+1/2]×[yj-1/2, yj+1/2], wherein subscript i, j are coordinate serial number.
Define UI, jRepresent U in grid cell II, jInterior value, i.e. UI, j=U (xi, yj)。tnRepresent the n-th time horizon, Δ t tables Show time step, then tn+1=tn+Δt。Represent UI, jIn the value of the n-th time horizon, following process entirely the n-th layer time into Row, so temporarily time variable t is not considered, it willIt is abbreviated as UI, j。
First, for above-mentioned shallow water equation group, constructing linear power can appoint the finite difference WENO forms taken discrete logical The derivative term about space variable is measured, detailed process is as follows：
1st step, to the space derivation item F (U) in equation groupxWith G (U)yCarrying out difference approximation has：
Wherein, (xi, yj) represent serial number i, j point coordinate,X directions and y directions are represented respectively Point (xi±1/2, yj) and (xi, yj±1/2) at numerical flux.
2nd step, to each coordinate points (xi, yj) at flux carry out Lax-Friedrichs thinking positions：
F(UI, j)=F (UI, j)++F(UI, j)-, (5)
G(UI, j)=G (UI, j)++G(UI, j)-, (6)
Wherein, UI, jRepresent U in point (xi, yj) at functional value, αlRepresent the maximum eigenvalue of x directions flux matrix, αk Represent the maximum eigenvalue of y directions flux matrix.
3rd step passes through above-mentioned F (UI, j)±The linear power of value construction can appoint the new finite difference WENO forms taken, obtain F (UI+1/2, j)±The high-order approximation value at place, it is similar to pass through G (UI, j)±Be worth to G (UI+1/2, j)±Value, specific steps are such as Under：
Step 3.1 first fixes y=yj, take the template T of 5 unit spots0={ II-2, j, II-1, j, II, j, II+1, j, II+2, j, structure Make a quartic polynomialIt is another to take 2 template T for including 2 unit spots1={ II-1, j, II, jAnd T2={ II, j, II+1, j, construct two linear polynomialsWithMeet following constraints respectively：
For convenience of statement, by F±(UL, j) be denoted asThenExpression be：
Step 3.2, which is appointed, takes one group and the positive number γ for 10, γ1, γ2As linear power.Compared to original finite difference WENO forms, linear power here can arbitrarily value without influencing form in 5 rank precision of smooth domain rather than unique true The fixed and positive number related with grid.
The multinomial that step 3.3 goes out according to three structure of transvers plateCalculate smooth indicator
Wherein, n represents corresponding polynomial serial number, and α represents summing target, and r represents corresponding polynomial number,Representative polynomialα partial derivative is asked to independent variable x.
Further,Expression be：
Step 3.4 calculates nonlinear weight according to the linear power of arbitrary value and smooth indicatorIt calculates public Formula is as follows：
The nonlinear weight that the multinomial and step 3.4 that step 3.5 is obtained according to step 3.1 obtain, obtains final F (UI+1/2, j)±High-order approximation value：
Step 3.6 fixes x=x againi, step 3.1-3.5 is repeated, it is similar to can obtain G (UI, j+1/2)±High-order approximation value.
Secondly, the influence for reduction riverbed topography logarithm form, the well-balanced technologies of use, concrete operations It is as follows：
1st step, the riverbed effect item S divided in shallow water equation group are as follows：
2nd step solves two source item S after division with above-mentioned improved finite difference WENO1, S2；
3rd step, and then obtain half Discrete Finite difference scheme of flow cavitation result equation：
Wherein, L (UI,J)=- F (U)x-G(U)y+S1+S2, represent to obtain after solving by improved finite difference WENO forms The value arrived.
Finally, using the discrete half Discrete Finite difference scheme of Runge-Kutta time discrete methods for meeting TVD properties, It is as follows so as to obtain space-time approximate shceme finite difference scheme：
Wherein,WithTo calculate intermediate transition value.Direction in space and time orientation it is discrete after the completion of obtain Space-time approximate shceme finite difference scheme is the iterative formula about time horizon, it is known that depth of water h, the x directions of initial streamflow regime The water velocity v in water velocity u, y directions, then constantly recycles according to iterative formula, you can obtain flow at a time or The high-precision numerical approximation value of physical quantity during stable state.
Specific embodiment of four examples as presently disclosed method is given below.
Embodiment one, the water surface perturbation problem in smooth riverbed.
Wave of Two-Dimension Shallow Water Equation is solved in region [0,2] × [0,1], riverbed is an elliptoid hump, as follows：
The depth of water and flow of original state is respectively：
Hu (x, y, 0)=hv (x, y, 0)=0.
It adds some points in the region x ∈ [0.05,0.15] microvariations, the tranquil water surface can form fluctuation and turn right propagation.Using 200 × 100 calculating grid, it is respectively t=0.12s, 0.24s, 0.36s, 0.48s, 0.60s to calculate the time.Fig. 1 is riverbed and river Flow profile, Fig. 2 a to Fig. 2 e depict the water surface vertical view calculated with the method for the present invention at each moment, apparent from figure The design sketch in propagation and process hump riverbed it can be seen that disturbance is turned right.
Embodiment two, the water surface perturbation problem for being interrupted riverbed.
Wave of Two-Dimension Shallow Water Equation is solved in region [0,2] × [0,1], riverbed is a hump for including interruption, as follows：
The depth of water and flow of original state is respectively：
Hu (x, y, 0)=hv (x, y, 0)=0.
In the interior microvariations of adding some points of region x ∈ [0.05,0.15], the tranquil water surface can form fluctuation and turn right between propagation and process Disconnected hump riverbed.We use 200 × 100 calculating grid, and it is respectively t=0.30s, 0.45s to calculate the time.Fig. 3 is drawn The topographic map in riverbed, Fig. 4 a to Fig. 4 b depict the water surface vertical view calculated with the method for the present invention at each moment, from figure In the significantly propagation and by the design sketch in hump riverbed it can be seen that disturbance is turned right.
Embodiment three, flat riverbed cylinder Dam Break Problems.
There is the round dam that Radius is 11m in zoning [0,50m] × [0,50m], the depth of water in dam is 10m, The outer depth of water 1m of dam.Since artificial or natural cause leads to dam body moment avalanche, lead to the water in dam rapidly toward external diffusion.Fortune This process is simulated with the method for the present invention.It is 200 × 200 to calculate grid, and it is respectively t=0.0s, 0.2s, 0.4s to calculate the time, 0.6s, 0.8s, 1.0s.Fig. 5 a to Fig. 5 f feature the process of this round Dam Break Problems.
The cylinder Dam Break Problems of example IV, hump riverbed.
This problem describes round Dam Break Problems of bottom tool there are one hump shape riverbed, zoning for [0, 2] × [0,2], it is 200 × 200 to calculate grid, and the calculating time is t=0.15s.The expression formula in riverbed is as follows：
The depth of water and flow of original state is respectively：
Hu (x, y, 0)=hv (x, y, 0)=0.0.
Fig. 6 depicts the depth of water of end time and the result figure in riverbed.
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 the protection of the present invention Range.

Claims (7)

  1. A kind of 1. Flow Numerical Simulation method based on improved finite difference scheme, which is characterized in that establish space right-angle seat Mark system, with uniform grid subdivision zoning, in the calculating grid of formation, the basic weighting dead-beat lattice of 5 rank precision of construction Formula carries out the Digital calculation modelling in zoning, specifically includes following steps：
    Step 1: in the numerical computations region of waters, generate rectangular coordinate system and carry out mesh generation；
    Step 2: the discrete shallow water equation of finite difference WENO forms with 5 rank precision is constructed on the calculating grid of formation Space derivation item in group, the linear power in construction process can arbitrary value；
    Step 3: the variation of circulation and riverbed gradient is balanced using well-balanced methods；
    Step 4: pushing away on time orientation is carried out to shallow water equation group using classical Runge-Kutta time discretes method Into so as to obtain the flow depth of any time, the speed of all directions and water flow, so as to which Numerical-Mode draws up practical problem In each moment streamflow regime.
  2. 2. a kind of Flow Numerical Simulation method based on improved finite difference scheme as described in claim 1, feature exist In：In the step 1, carry out the respective grid points coordinate after mesh generation and be denoted as (xi, yj), wherein x, y representation space variables, Subscript i, j represent grid serial number.
  3. 3. a kind of Flow Numerical Simulation method based on improved finite difference scheme as claimed in claim 2, feature exist In：In the step 2, the shallow water equation group for describing water movement is：
    Above-mentioned equation group is uniformly write as：Ut+F(U)x+G(U)y=S, wherein, t expression time variables, x, y representation space variables, U=(h, hu, hv)TRepresent conservation variable vector, F (U), G (U) represents flux vector, F (U)xRepresent F (U) to x derivations, G (U)yRepresent G (U) to y derivations, S=(0 ,-ghbx,- ghby)TRepresent riverbed effect item, h, u, v, b are the functions about time variable t and space variable x, y, represent flow respectively Depth, horizontal direction speed, vertical direction speed and bed level, bxRepresent b to x derivations, byB is represented to y derivations, T is represented Transposition, g represent acceleration of gravity.
  4. 4. a kind of Flow Numerical Simulation method based on improved finite difference scheme as claimed in claim 3, feature exist In：For the shallow water equation group, linear power can appoint the construction process of finite difference WENO forms taken as follows：
    1st step, to the space derivation item F (U) in equation groupxWith G (U)yCarrying out difference approximation has：
    Wherein, (xi, yj) represent serial number i, j point coordinate,X directions and y directions point are represented respectively (xi±1/2, yj) and (xi, yj±1/2) at numerical flux；
    2nd step, to each coordinate points (xi, yj) at flux carry out Lax-Friedrichs thinking positions：
    F(UI, j)=F (UI, j)++F(UI, j)-,
    G(UI, j)=G (UI, j)++G(UI, j)-,
    Wherein, UI, jRepresent U in point (xi, yj) at functional value, αlRepresent the maximum eigenvalue of x directions flux matrix, αkRepresent y The maximum eigenvalue of direction flux matrix；
    3rd step passes through F (UI, j)±The linear power of value construction can appoint the new finite difference WENO forms taken, obtain F (UI+1/2, j)± The high-order approximation value at place similarly, passes through G (UI, j)±Be worth to G (UI+1/2, j)±Value.
  5. 5. a kind of Flow Numerical Simulation method based on improved finite difference scheme as claimed in claim 4, feature exist In：3rd step specifically comprises the following steps：
    Step 3.1 first fixes y=yj, take the template T of 5 unit spots0={ II-2, j, II-1, j, II, j, II+1, j, II+2, j, construction one A quartic polynomialIt is another to take 2 template T for including 2 unit spots1={ II-1, j, II, jAnd T2={ II, j, II+1, j, construct two linear polynomialsWithMeet following constraints respectively：
    By F±(UI, j) be denoted asThenExpression be：
    Step 3.2 appoints the positive number γ for taking one group and being 10, γ1, γ2As linear power；
    Step 3.3, the multinomial gone out according to three structure of transvers plateCalculate smooth indicator
    Wherein, n represents corresponding polynomial serial number, and α represents summing target, and r represents corresponding polynomial number, Representative polynomialα partial derivative is asked to independent variable x；
    ThenExpression be：
    Step 3.4, linearly power and smooth indicator according to arbitrary value, calculate nonlinear weightCalculation formula It is as follows：
    Wherein,It represents to calculate transition value；
    The nonlinear weight that step 3.5, the linear term formula obtained according to step 3.1 and step 3.4 obtain, obtains final F (UI+1/2, j )±High-order approximation value：
    Step 3.6 fixes x=x againi, obtain G (UI, j+1/2)±High-order approximation value.
  6. 6. a kind of Flow Numerical Simulation method based on improved finite difference scheme as claimed in claim 4, feature exist In：In the step 3, it is as follows using well-balanced methods：
    1st step, the riverbed effect item S divided in shallow water equation group are as follows：
    Wherein, S1, S2Represent two source items after division；
    2nd step solves two source item S after division with finite difference WENO1, S2；
    3rd step obtains half Discrete Finite difference scheme of flow cavitation result equation：
    Wherein L (UI, j)=- F (U)x-G(U)y+S1+S2, represent the high-order approximation obtained after being solved by finite difference WENO forms Value.
  7. 7. a kind of Flow Numerical Simulation method based on improved finite difference scheme as claimed in claim 6, feature exist In：In the step 4, time step is Δ t, tnRepresent the n-th time horizon,Represent U (xi, yj, tn) value, in tnTime horizon It is calculatedThen discrete half Discrete Finite of Runge-Kntta time discrete methods for meeting TVD properties is utilized Difference scheme is as follows so as to obtain space-time approximate shceme finite difference scheme：
    Wherein,WithTo calculate the discrete space-time obtained after the completion of intermediate transition value, direction in space and time orientation Approximate shceme finite difference scheme is the iterative formula about time horizon, it is known that the flow in depth of water h, the x directions of initial streamflow regime The water velocity v in speed u, y directions, then constantly recycles according to iterative formula, you can obtains flow at a time or stablizes The high-precision numerical approximation value of physical quantity during state.
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