Second-order precision gradient adaptive terrain water level reconstruction method based on dimensionless wavenumber

Abstract

The invention provides a second-order precision gradient adaptive terrain water level reconstruction method based on a dimensionless wavenumber. The method comprises the following steps: acquiring a one-dimensional hydrodynamic model and a two-dimensional hydrodynamic model; dividing dry and wet types according to the dry and wet states of the central grid and the critical grid of the central grid; for the center grids of different dry and wet types, according to the terrain water level information of the center grids and the critical grids, the terrain water level of the center grids is preliminarily reconstructed; correcting the preliminarily reconstructed terrain water level according to a terrain water level gradient method for a central wet grid or a dry grid in a dry and wet boundary; using a second-order precision discontinuous detector to judge whether the central grid and the critical grid are in a discontinuous region or not; if the central grid is located in the discontinuous region, performing equivalent dimensionless wavenumber calculation on interfaces on the front side and the rear side of the central grid in the forward direction of the calculation dimension, and further obtaining the self-adaptive dissipation amount of the central grid; and normalizing the self-adaptive dissipation amount to obtain an adjustment proportion of the terrain water level gradient, and adjusting the gradient in the terrain water level gradient method to complete terrain water level reconstruction of the self-adaptive gradient.

Description

Technical Field

[0001] This invention relates to the technical field of watershed hydrological modeling, and in particular to a second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber. Background Technology

[0002] One- or two-dimensional hydrodynamic models based on the Godunov-type finite volume method require source term handling during computation. Source term handling is a crucial step in ensuring model accuracy and stability, especially the handling of the bottom slope term, which is particularly important in simulating water flow behavior in complex terrain. Hydrostatic reconstruction is a numerical technique used to handle the bottom slope term, primarily addressing imbalances in shallow water equations. Its main purpose is to maintain hydrostatic equilibrium in numerical simulations, thereby avoiding spurious flows or numerical errors introduced due to improper handling of the bottom slope term.

[0003] Among existing research, the topographic water level gradient reconstruction method is a relatively advanced still water reconstruction method. This method exhibits good balance and reliability, and avoids non-physical reflections near topographic steps. However, it sometimes leads to numerical instability when dealing with complex terrain problems. Therefore, this method still has certain limitations in handling complex terrain issues, and further optimization of model design is urgently needed to improve its robustness. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumbers. This method uses topographic water level gradient reconstruction and flow field variation information near complex terrain to determine the adaptive adjustment ratio by calculating the corresponding equivalent dimensionless wavenumbers, thereby correcting the gradient used to reconstruct the topographic water level and thus completing the adaptive reconstruction of the topographic water level.

[0005] The purpose of this invention is to provide a second-order accuracy gradient adaptive topographic water level reconstruction method based on dimensionless wavenumbers, including obtaining a hydrodynamic model, and further including the following steps:

[0006] Step 1: In the hydrodynamic model, the dry / wet types of the central grid are classified according to the dry / wet state of the central grid itself and its critical grids;

[0007] Step 2: For the central grids of different wet and dry types, calculate the reconstruction gradient based on the topographic and water level information of the grids themselves and the critical grids and a specific slope limiter, and perform preliminary reconstruction of the topographic and water level of the central grids;

[0008] Step 3: For the central grid that is a wet grid or a dry grid located at the wet-dry boundary, the topographic water level of the preliminary reconstruction is corrected according to the topographic water level gradient method;

[0009] Step 4: Using a second-order precision discontinuity detector, detect the smoothness of the water depth change near the center and critical grid to determine whether the center and critical grid are in the discontinuity zone.

[0010] Step 5: If the central or critical grid is in a discontinuity region, calculate the equivalent dimensionless wavenumber of the central grid on both the front and back interfaces in the positive direction of the computation dimension;

[0011] Step 6: Based on the equivalent dimensionless wavenumber, the adaptive dissipation of the central grid is calculated using the relationship between dissipation and equivalent dimensionless wavenumber.

[0012] Step 7: Normalize the adaptive dissipation to obtain the adjustment ratio θ of the terrain water level gradient;

[0013] Step 8: Using the adjustment ratio θ, adjust the gradient in the topographic water level gradient method to complete the adaptive gradient topographic water level reconstruction of the hydrodynamic model.

[0014] Preferably, the classification of dry and wet types includes:

[0015] set up Given a one-dimensional region, discretize the region as Ω=U i∈I C i , among which, U i∈I For global grid, grid and in the set of integers As an index set For grid The boundary is calculated on the positive backward side of the dimension. For grid The boundary on the positive front side of the calculated dimension;

[0016] Unify all grids, so that ∆x is the grid side length;

[0017] Let b i =b i (t) and η i =η i Let (t) represent the average topography and water level within the i-th grid at time t, then h i =h i (t)=η i (t)-b i (t) represents the water depth corresponding to the i-th grid, b i η i and h i All values ​​are assigned to the center of the grid.

[0018] In any of the above schemes, it is preferred that the grid type be classified as wet grid I based on the dryness or wetness state inside the central grid. wet (t) and dry grid I dry (t) Two types, the wet mesh I wet (t) is

[0019] I wet (t):={i∈I|h i >0}

[0020] The dry grid I dry (t) is

[0021] I dry (t):={i∈I|h i =0};

[0022] Considering the wet / dry state of the critical mesh in the central mesh, the central mesh is divided into dry meshes located at the wet / dry boundary. Wet grid at the wet-dry boundary Wet mesh inside a moist environment and the dry mesh inside the dry area Four types,

[0023] The dry grid at the dry-wet boundary for

[0024]

[0025] The wet grid at the dry-wet boundary for

[0026]

[0027] The wet mesh inside the wet area for

[0028]

[0029] The dry mesh located inside the dry area for

[0030]

[0031] Wherein, min(h) i±1 ) represents the minimum water depth in the grids before and after the central grid's calculation dimension, max(h) i±1 The maximum water depth is calculated in the grids on both sides of the central grid.

[0032] In any of the above schemes, the preliminary reconstruction preferably includes:

[0033] For the variable q∈(η,b), in the central grid C i The initial linear reconstruction relation within:

[0034]

[0035] in, It is the central grid C i The reconstructed gradient of variable q, where x is the coordinate of each point in the positive direction of the computation dimension within the grid, and q i (t) represents the value of the grid center reconstruction variable, x i The coordinates of the center point inside the grid;

[0036] If the central grid belongs to Configurable settings:

[0037]

[0038] If the central grid belongs to or I wet (t), using the minmod slope limiter to compute the gradient for the initial reconstruction of the variable q∈(η,b):

[0039]

[0040] The minmod function is:

[0041]

[0042] Where θ typically takes values ​​in the range [1.0, 2.0], q i-1 Reconstruct variable values ​​for the meshes behind the central mesh, q i Reconstruct variable values ​​for the central grid. Let q be the grid side length. i+1 Here, a1 represents the reconstructed variable value of the grid in front of the central grid, and a1 is the first slope estimate to be compared. m Let m be the m-th slope estimate to be compared, where m is the number of slope estimates to be compared.

[0043] In any of the above schemes, it is preferable to introduce a one-sided limit at the boundary of the central grid, in the central grid C. i boundary At this point, the left limit is:

[0044]

[0045] The right-hand limit is:

[0046] .

[0047] In any of the above schemes, step 3 preferably includes reconstruction correction based on the topographic water level gradient method, including:

[0048] (1) When it appears hour:

[0049]

[0050] For variables q∈(η,b), the gradient used to reconstruct the topographic water level is adjusted, denoted as:

[0051]

[0052] (2) When it appears hour:

[0053]

[0054] For variables The gradient used to reconstruct the topographic water level is adjusted, denoted as:

[0055]

[0056] Based on the readjusted gradient It can be applied to the central grid C. i The topographic water level is reconstructed and corrected according to the following formula:

[0057]

[0058] in, The water level is on the right side of the rear boundary of the central grid. The terrain to the right of the rear boundary of the central grid. Let t be the water level on the right side of the rear boundary of the central grid. Let t be the water level at the center grid. The terrain to the right of the rear boundary of the central grid at time t; The water level is on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid. Let t be the water level on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid at time t; Correction values ​​for the topographic water level reconstruction gradient.

[0059] In any of the above solutions, the preferred option is when the following occurs: hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections;

[0060] When it appears or or hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient Make corrections;

[0061] When it appears hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections;

[0062] When it appears hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections;

[0063] When it appears hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections;

[0064] When it appears hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient All have been revised;

[0065] When it appears or hour, For the preliminary reconstructed terrain gradient and the initially reconstructed water level gradient All have been revised;

[0066] When it appears hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections.

[0067] in, This is the reconstruction correction value inside the rear boundary of the central mesh. This is the reconstruction correction value inside the front boundary of the central grid.

[0068] In any of the above embodiments, the preferred method for operating the intermittent detector includes:

[0069] Step 41: For water depth information h i hi+2 h i+3 h i+1 h i-1 and h i-2 After normalization, the formula is: denom = max(1.0, max(h) i ,h i+2 ,h i+3 ,h i+1 ,h i-1 ,h i-2 )),

[0070] Normalization of water depth information:

[0071] h k1 =h k1 / denom;

[0072] Step 42: Calculate the detection center grid C of the intermittent detector. i discontinuity ψ i The formula is

[0073]

[0074] a=|h i -h i-1 |+|h i -2h i-1 +h i-2 |

[0075] a=|h i -h i+1 |+|h i -2h i+1 +h i+2 |

[0076]

[0077] Step 43: When ψ i ≥ψ c If the water depth change near the detection grid is smooth, the detection grid is considered to be in a discontinuous zone.

[0078] Among them, k1={i-2,i-1,i,i+1,i+2,i+3}, ξ 2 =10 2 , ψ c It is a constant.

[0079] In any of the above solutions, step 5 preferably includes the following sub-steps:

[0080] Step 51: When the center grid C i To reconstruct and correct the mesh and min(ψ) i-1 ,ψ i,ψ i+1 )≥ψ c At that time, the water depth information h i+1 h i+2 h i+3 h i-2 h i-1 and h i Normalization is performed, and the following definition is made:

[0081] denom=max(1.0,max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i )),

[0082] h max =max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / denom,

[0083] h min =min(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / denom,

[0084] Normalization of water depth information:

[0085] h k2 =h k2 / denom;

[0086] Step 52: Calculate the dimension of the central network forward and backward on both sides of the interface. and Calculate the equivalent dimensionless wavenumber and The formula is

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097]

[0098] Among them, k2={i+2,i+1,i,i-2,i-1,i-3}.

[0099] In any of the above schemes, step 6 preferably includes determining the equivalent dimensionless wavenumber. and The maximum value in Is it higher than the critical wavenumber k? c ,

[0100] If k ESW Below k c If the reconstructed correction mesh is in a smooth region, then the dissipation γ of the reconstructed correction mesh is considered to be within the smooth region. diss Set to 0.001;

[0101] If k ESW Higher than k c Then the dissipation γ is used. diss With equivalent dimensionless wavenumber k ESW The dissipation of the reconstructed and corrected mesh is calculated using a specific relation, as follows:

[0102]

[0103] Where, k max =π.

[0104] In any of the above schemes, it is preferred that the adjustment ratio θ of the topographic water level gradient is related to the dissipation γ. diss The relationship is as follows:

[0105] .

[0106] Preferably, in any of the above schemes, the adaptive gradient topographic water level reconstruction includes the following cases:

[0107] Scenario 1: When At that time, the topographic water level gradient method was used to obtain... and , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0108]

[0109]

[0110] Scenario 2: When At that time, the topographic water level gradient method was used to obtain... , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0111]

[0112]

[0113] Scenario 3: When At that time, the topographic water level gradient method was used to obtain... , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0114]

[0115]

[0116] Scenario 4: When At that time, the topographic water level gradient method was used to obtain... , , For positive, Negative,

[0117] when hour, , ,

[0118] when hour; , ;

[0119] Scenario 5: When At that time, the topographic water level gradient method was used to obtain... , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Keeping the topography and water level unchanged, the adaptive reconstruction gradients are as follows:

[0120]

[0121]

[0122] Scenario 6: When At that time, the topographic water level gradient method was used to obtain... , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0123]

[0124]

[0125] Scenario 7: When At that time, the topographic water level gradient method was used to obtain... , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0126]

[0127]

[0128] Situation 8: When At that time, the topographic water level gradient method was used to obtain... , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0129]

[0130]

[0131] Situation 9: When At that time, the topographic water level gradient method was used to obtain... , , Negative, For positive,

[0132] when hour

[0133]

[0134]

[0135] when hour:

[0136]

[0137]

[0138] Case 10: When At that time, the topographic water level gradient method was used to obtain... , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Remaining constant, the adaptive reconstruction gradients for topography and water level are as follows:

[0139]

[0140] .

[0141] In any of the above schemes, it is preferred to calculate the reconstructed correction mesh AR. i At time t, the topography and water depth q*(x,t) (q∈(η,b) at each point along the dimensional direction are calculated to complete the adaptive reconstruction of the topography and water level. The adaptive reconstruction relationship is as follows:

[0142] .

[0143] This invention proposes a second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber, which solves the numerical instability problem exhibited by the topographic water level gradient reconstruction method when dealing with complex terrain. It has the advantages of strict mass conservation, harmony, positive preservation and robustness, and has important academic significance and application value. Attached Figure Description

[0144] Figure 1 This is a flowchart of a preferred embodiment of the second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber according to the present invention.

[0145] Figure 2 To correct the AR (Adaptive Terrain and Water Level Reconstruction) mesh according to the dimensionless wavenumber-based second-order precision gradient adaptive topography and water level reconstruction method of the present invention. i A schematic diagram of an embodiment of the topographic water level gradient method reconstruction correction at time t.

[0146] Figure 3 To correct the AR (Adaptive Terrain and Water Level Reconstruction) mesh according to the dimensionless wavenumber-based second-order precision gradient adaptive topography and water level reconstruction method of the present invention. i At time t and at A schematic diagram of an embodiment of adaptive gradient terrain water level reconstruction correction.

[0147] Figure 4 To correct the AR (Adaptive Terrain and Water Level Reconstruction) mesh according to the dimensionless wavenumber-based second-order precision gradient adaptive topography and water level reconstruction method of the present invention. i At time t and at A schematic diagram of an embodiment of adaptive gradient terrain water level reconstruction correction. Detailed Implementation

[0148] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0149] Example 1

[0150] Based on the one- or two-dimensional hydrodynamic model of the Godunov-type finite volume method, this invention provides a second-order accuracy gradient adaptive topographic water level reconstruction method based on dimensionless wavenumbers, such as... Figure 1 As shown, it includes the following steps:

[0151] Perform step 100 to obtain one-dimensional and two-dimensional hydrodynamic models.

[0152] Execute step 110 to classify the dryness / wetness type of the central grid based on the dryness / wetness state of the central grid itself and its critical grids.

[0153] 1. Classification of dry and wet types in the central grid:

[0154] Taking a one-dimensional hydrodynamic model as an example, let... Given a one-dimensional region, discretize the region as Ω=U i∈I C i , among which, U i∈I For global grid, grid and in the set of integers As an index set For grid The boundary is calculated on the positive backward side of the dimension. For grid At the boundary directly ahead of the calculated dimension; unify all meshes, making... ∆x is the grid side length.

[0155] Let b i =b i (t) and η i=η i Let (t) represent the average topography and water level within the i-th grid at time t, then h i =h i (t)=η i (t)-b i (t) represents the water depth corresponding to the i-th grid, b i η i and h i All values ​​are assigned to the center of the grid.

[0156] Based on the wet / dry state of the central grid, the grid type can be divided into wet grid I. wet (t) and dry grid I dry (t) Two types, wet mesh I wet (t) is

[0157] I wet (t):={ i∈I|h i >0}

[0158] Dry Mesh I dry (t) is

[0159] I dry (t):={ i∈I|h i =0};

[0160] Simultaneously considering the wet / dry state of the critical grid of the central grid, the central grid is divided into dry grids located at the wet / dry boundary. Wet grid at the wet-dry boundary Wet mesh inside a moist environment and the dry mesh inside the dry area Four types,

[0161] The dry grid at the dry-wet boundary for

[0162]

[0163] The wet grid at the dry-wet boundary for

[0164]

[0165] The wet mesh inside the wet area for

[0166]

[0167] The dry mesh located inside the dry area for

[0168]

[0169] Wherein, min(h) i±1 ) represents the minimum water depth in the grids before and after the central grid's calculation dimension, max(h) i±1 The maximum water depth is calculated in the grids on both sides of the central grid.

[0170] Step 120 involves calculating a reconstruction gradient for the central grids of different wet / dry conditions based on their own topographic and water level information and the critical grids, along with a specific slope limiter. This initial reconstruction of the topographic and water level of the central grids includes:

[0171] For the variable q∈(η,b), in the central grid C i Internally define a preliminary linear reconstruction relation:

[0172]

[0173] in, It is the central grid C i The reconstructed gradient of variable q, where x is the coordinate of each point in the positive direction of the computation dimension within the grid, and q i (t) represents the value of the grid center reconstruction variable, x i The coordinates of the center point inside the grid;

[0174] If the central grid belongs to Configurable settings:

[0175]

[0176] If the central grid belongs to or I wet (t), using the minmod slope limiter to compute the gradient for the initial reconstruction of the variable q∈(η,b):

[0177]

[0178] The minmod function is:

[0179]

[0180] The parameter θ controls the numerical viscosity of the numerical format, and the value range of θ is usually set to [1.0, 2.0]. i-1 Reconstruct variable values ​​for the grid behind the central grid, q i Reconstruct variable values ​​for the central grid. Let q be the grid side length. i+1 Here, a1 represents the reconstructed variable value of the grid in front of the central grid, and a1 is the first slope estimate to be compared. mLet m be the m-th slope estimate to be compared, where m is the number of slope estimates to be compared.

[0181] Determine variables gradient Then, by substituting these values ​​into the linear reconstruction formula, the central grid can be completed. The topography and water level have been preliminarily reconstructed.

[0182] To facilitate subsequent reconstruction and correction, a one-sided limit is introduced at the boundary of the central grid. In the central grid C... i boundary At this point, the left limit is:

[0183]

[0184] The right-hand limit is:

[0185] .

[0186] Step 130 is executed, where for a central grid that is a wet grid or a dry grid located at the wet-dry boundary, the topographic water level of the initially reconstructed grid is corrected according to the topographic water level gradient method.

[0187] For type or central grid There is a possibility that the terrain inside the boundary may be higher than the water level, which is an unreasonable phenomenon. or The central mesh where this phenomenon occurs is called the "reconstruction correction mesh," denoted as... The initially reconstructed topographic water level needs to be corrected again, that is, the gradient required for topographic water level reconstruction needs to be adjusted. This correction method is called the topographic water level gradient method, and its implementation is as follows:

[0188] because and These two situations cannot occur simultaneously, therefore they must be handled separately:

[0189] (1) When it appears hour:

[0190]

[0191] For variables q∈(η,b), the gradient used to reconstruct the topographic water level is adjusted, denoted as:

[0192]

[0193] (2) When it appears hour:

[0194]

[0195] For variables The gradient used to reconstruct the topographic water level is adjusted, denoted as:

[0196]

[0197] Based on the readjusted gradient It can be applied to the central grid C. i The topographic water level is reconstructed and corrected according to the following formula:

[0198]

[0199] in, The water level is on the right side of the rear boundary of the central grid. The terrain to the right of the rear boundary of the central grid. Let t be the water level on the right side of the rear boundary of the central grid. Let t be the water level at the center grid. The terrain to the right of the rear boundary of the central grid at time t; The water level is on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid. Let t be the water level on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid at time t; Correction values ​​for the topographic water level reconstruction gradient.

[0200] like Figure 2 The image shown is of the "Reconstructed Corrected Mesh". A schematic diagram of topographic water level gradient reconstruction correction at time t. The gray dots are connected by lines representing... Parallel auxiliary lines; red dashed lines represent the initially reconstructed terrain. The blue dashed line represents the initially reconstructed water level. The red solid line represents the reconstructed and corrected terrain. The blue solid line represents the reconstructed and corrected water level. ; Figure 2 The red circles in (a), 2(b), and 2(c) are , Figure 2 The red circles in (d), 2(e), and 2(f) are .

[0201] (1) Figure 2 (a) indicates when it occurs hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections;

[0202] (2) Figure 2 (b) indicates when it occurs or or hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient Make corrections;

[0203] (3) Figure 2 (c) indicates that when it occurs hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections;

[0204] (4) Figure 2 (d) indicates when it occurs hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections;

[0205] (5) Figure 2 (e) indicates that when it occurs or or hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient All have been revised;

[0206] (6) Figure 2 (f) indicates when the occurrence hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections.

[0207] in, This is the reconstruction correction value inside the rear boundary of the central mesh. This is the reconstruction correction value inside the front boundary of the central grid.

[0208] Step 140 involves using a second-order precision discontinuity detector to assess the smoothness of water depth changes near the center and critical grid, determining whether the center and critical grid are in discontinuity zones. The detection principle of the discontinuity detector is as follows:

[0209] For water depth information h i h i+2 h i+3 h i+1 hi-1 and h i-2 After normalization, the formula is: denom = max(1.0, max(h) i ,h i+2 ,h i+3 ,h i+1 ,h i-1 ,h i-2 )),

[0210] Normalization of water depth information:

[0211] h k1 = h k1 / denom;

[0212] Among them, k= i-2, i -1, i, i +1, i +2, i +3.

[0213] Intermittent detector detects the central grid C i discontinuity ψ i The formula is as follows

[0214]

[0215] a=|h i - h i-1 |+|h i -2h i-1 + h i-2 |

[0216] a=|h i - h i+1 |+|h i -2h i+1 + h i+2 |

[0217]

[0218] Where, ξ 2 =10 2 , ψ c As a constant, when ψ i ≥ψ c If the water depth change near the detection grid is smooth, it is considered that the detection grid is discontinuous.

[0219] Execute step 150. If the central or critical grid is in a discontinuity region, calculate the equivalent dimensionless wavenumber of the central grid on the front and back interfaces in the computational dimension.

[0220] If at least one of the "reconstructed correction grid" and its critical grid is in a discontinuity region, then the equivalent dimensionless wavenumber of the "reconstructed correction grid" is calculated on both the front and back interfaces in the computational dimension. Specifically, the calculation of the equivalent dimensionless wavenumber of a certain interface of the "reconstructed correction grid" utilizes the water depth information from six grids: three grids on the front side and three grids on the back side in the normal direction of that interface.

[0221] When the central grid For "reconstructing and correcting the mesh" and min( )≥ At this time, it is necessary to calculate its dimension forward and backward on both sides of the interface. and Calculate the equivalent dimensionless wavenumber, denoted as follows: and . To calculate Taking the interface as an example, the water depth information used is , , , , and .

[0222] When the center grid C i To reconstruct and correct the mesh and min(ψ) i-1 , ψ i , ψ i+1 )≥ψ c At this time, it is necessary to calculate its dimension forward and backward on both sides of the interface. and Calculate the equivalent dimensionless wavenumber, denoted as follows: and . To calculate Taking the interface as an example, the water depth information used is h. i+1 h i+2 h i+3 h i-2 h i-2 and h i .

[0223] First, still regarding the water depth information h i+1 h i+2 h i+3 h i-2 h i-1 and h i Normalization is performed, and the following definition is made:

[0224] denom=max(1.0,max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,hi )),

[0225] h max = max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / denom,

[0226] h min = min(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / denom,

[0227] Normalization of water depth information:

[0228] h k2 = h k2 / denom;

[0229] The central network computes dimensions forward and backward across both interfaces. and Calculate the equivalent dimensionless wavenumber and The formula is

[0230]

[0231]

[0232]

[0233]

[0234]

[0235]

[0236]

[0237]

[0238]

[0239]

[0240]

[0241] Among them, k2={i +2, i +1, i, i-2, i -1, i-3}.

[0242] Execute step 160: Based on the equivalent dimensionless wavenumber, calculate the adaptive dissipation of the central grid using the relationship between dissipation and equivalent dimensionless wavenumber; determine the equivalent dimensionless wavenumber. and The maximum value in Is it higher than the critical wavenumber k? c ,

[0243] If k ESW Below k c If the reconstructed correction mesh is in a smooth region, then the dissipation γ of the reconstructed correction mesh is considered to be within the smooth region. diss Set to 0.001;

[0244] If k ESW Higher than k c Then the dissipation γ is used. diss With equivalent dimensionless wavenumber k ESW The dissipation of the "reconstructed correction mesh" is calculated using a specific relation, as follows:

[0245]

[0246] Where, k max =π.

[0247] Step 170 involves linearly normalizing the adaptive dissipation calculated in step 160 to obtain the adjustment ratio θ for the topographic water level reconstruction gradient. A smaller dissipation indicates a smoother water depth flow field near the "reconstructed correction grid," requiring a smaller adjustment ratio; a larger dissipation indicates a greater discontinuity in the water depth flow field near the "reconstructed correction grid," requiring a larger adjustment ratio. The gradient adjustment ratio θ is related to the dissipation. The relationship is as follows:

[0248] .

[0249] Execute step 180, using the gradient adjustment scale generated in step 170, and combining it with the calculation results from the topographic water level gradient method in step 3, to reconstruct and correct the AR mesh. i The terrain reconstruction gradient belongs to and water level reconstruction gradient The gradient of the larger absolute value is linearly approximated to the smaller absolute value based on the gradient adjustment ratio, generating an adaptive reconstruction gradient for the terrain and water level. This is because the relative positions of the terrain and water level within the grid determine... and To determine which component has the larger absolute value, adaptive reconstruction is required in the following scenarios:

[0250] when At that time, gradient adjustment is performed in the following four cases, such as Figure 3 As shown (the meanings of the symbols and lines in the diagram are...) Figure 2 Consistent, the gray arrows indicate the directions of gradient adjustments that require adaptive adjustment, and the solid gray line represents the maximum extent to which the gradient can be adjusted (θ = 1.0):

[0251] (1) (like Figure 3 (a) shown)

[0252] Based on the topographic water level gradient method and , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0253]

[0254]

[0255] (2) When (like Figure 3 (b)

[0256] Based on the topographic water level gradient method , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0257]

[0258]

[0259] (3) When (like Figure 3 (c) shown)

[0260] Based on the topographic water level gradient method , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0261]

[0262]

[0263] (4)

[0264] According to the topographic water level gradient method, we can obtain... , , For positive, Since the values ​​are negative, it is impossible to directly determine the relationship between their absolute values; a comparison is required.

[0265] (like Figure 3 (d)

[0266] , ,

[0267] (like Figure 3 (e)

[0268] , ;

[0269] (5) (like Figure 3 (f)

[0270] Based on the topographic water level gradient method , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Keeping the topography and water level unchanged, the adaptive reconstruction gradients are as follows:

[0271]

[0272] .

[0273] when At that time, gradient adjustment is performed in the following four cases, such as Figure 4 As shown (the meanings of the symbols and lines in the diagram are...) Figure 1 Consistent, the gray arrows indicate the directions of gradient adjustments that require adaptive adjustment, and the solid gray line represents the maximum extent to which the gradient can be adjusted (θ = 1.0):

[0274] (6) (like Figure 4 (a) shown)

[0275] Based on the topographic water level gradient method , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0276]

[0277]

[0278] (7) (like Figure 4 (b)

[0279] Based on the topographic water level gradient method , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0280]

[0281]

[0282] (8) (like Figure 4 (c) shown)

[0283] Based on the topographic water level gradient method , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows:

[0284]

[0285]

[0286] (9) hour,

[0287] According to the topographic water level gradient method, we can obtain... , , Negative, If the result is positive, it is impossible to directly determine the relationship between the absolute values ​​of the two; a comparison is required.

[0288] like (like Figure 4 (d)

[0289]

[0290]

[0291] like (like Figure 4 (e)

[0292]

[0293]

[0294] (10) (like Figure 4 (f)

[0295] Based on the topographic water level gradient method , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Remaining constant, the adaptive reconstruction gradients for topography and water level are as follows:

[0296]

[0297] .

[0298] For variables In "Reconstructing and Correcting the Mesh" An adaptive reconstruction relation is defined internally:

[0299]

[0300] In the formula, This refers to "reconstructing and correcting the mesh". The adaptive reconstruction gradient of water level and terrain at time t is used to calculate the "reconstruction correction grid" using formula (37). At time t, calculate the terrain and water depth at various points along the dimensional direction. This allows for adaptive reconstruction of terrain and water levels.

[0301] Example 2

[0302] This invention proposes a second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber. Based on the topographic water level gradient reconstruction method and the flow field change information near complex terrain, the adaptive adjustment ratio is determined by calculating the corresponding equivalent dimensionless wavenumber to correct the gradient used to reconstruct the topographic water level, thereby completing the adaptive reconstruction of the topographic water level. This method can be used to solve the numerical instability problem exhibited by the topographic water level gradient reconstruction method when dealing with complex terrain.

[0303] A second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber includes the following steps:

[0304] Step 1: Classification of the dry and wet types of the central grid: Classify the dry and wet types of the central grid based on the dry and wet states of the central grid itself and its critical grids;

[0305] In step 1, the dryness / wetness of the central grid can be divided into four types based on its own dryness / wetness state and the dryness / wetness state of the critical grid.

[0306] Step 2, Preliminary Reconstruction of Topographic and Water Levels: For the central grids of different wet and dry types, the reconstruction gradient is calculated based on the topographic and water level information of the grids themselves and the critical grids and a specific slope limiter, so that the topographic and water levels of the central grids can be preliminarily reconstructed.

[0307] In step 2, if the type of the central grid is a wet grid or a dry grid at the wet-dry boundary, the gradient for the initial reconstruction of the topographic water level is calculated using the minmod slope limiter, and the initial reconstruction of the topographic water level is completed.

[0308] Step 3, Reconstruction and Correction Based on Topographic Water Level Gradient Method: For a wet grid at the center or a dry grid at the wet-dry boundary, the topographic water level initially reconstructed in Step 2 is corrected based on the topographic water level gradient method.

[0309] In step 3, for a central grid that is a wet grid or a dry grid at the wet-dry boundary, if an unreasonable phenomenon occurs where the topography is higher than the water level on the left and right sides after the initial reconstruction in step 2, the topography and water level of the grid are reconstructed and corrected according to the topography and water level gradient method, and the central grid is named "reconstruction correction grid".

[0310] Step 4: Calculation of discontinuity detector function value: Using a second-order precision discontinuity detector, the smoothness of the water depth change near the center and critical grid is detected to determine whether the center and critical grid are in the discontinuity zone.

[0311] In step 4, a second-order precision discontinuity detector is used to detect the smoothness of the water depth changes near the "reconstructed correction grid" and its critical grid. Then, based on the detected values, it is determined whether the "reconstructed correction grid" and its critical grid are in a discontinuity region. The information used when detecting the grid includes the water depth information of the grid itself, the two adjacent grids directly in front of it, and the two adjacent grids behind it, totaling five grids.

[0312] Step 5: Calculate the equivalent dimensionless wavenumber: If the central or critical grid is in a discontinuity region, calculate the equivalent dimensionless wavenumber of the central grid on both the front and back interfaces in the calculation dimension.

[0313] In step 5, if at least one of the "reconstructed correction grid" and its critical grid is in a discontinuity region, then the equivalent dimensionless wavenumber of the "reconstructed correction grid" is calculated on both the front and back interfaces in the computational dimension. Specifically, the calculation of the equivalent dimensionless wavenumber of a certain interface of the "reconstructed correction grid" utilizes the water depth information from six grids: three grids on the front side and three grids on the back side in the normal direction of that interface.

[0314] Step 6, Adaptive dissipation calculation: Based on the equivalent dimensionless wavenumber of the central grid on both sides of the positive front and back interfaces in the calculation dimension obtained in Step 5, the adaptive dissipation of the central grid can be calculated by using the relationship between dissipation and equivalent dimensionless wavenumber.

[0315] In step 6, it is first determined whether the maximum value of the equivalent dimensionless wavenumber of the “reconstructed correction mesh” obtained in step 5 on both sides of the positive front and back interfaces of the calculation dimension is higher than the critical wavenumber. If the value is lower than the critical wavenumber, it is considered that the “reconstructed correction mesh” is in the smooth region and the dissipation is very small, so the dissipation of the “reconstructed correction mesh” is directly set to 0.001. If the value is higher than the critical wavenumber, the dissipation of the “reconstructed correction mesh” is calculated using a specific relationship between dissipation and equivalent dimensionless wavenumber.

[0316] Step 7: Gradient adjustment ratio calculation: Normalize the adaptive dissipation calculated in Step 6 to obtain the adjustment ratio of the terrain water level gradient.

[0317] In step 7, the adaptive dissipation calculated in step 6 is linearly normalized to obtain the adjustment ratio of the topographic water level gradient. The smaller the dissipation, the smoother the water depth flow field near the "reconstructed correction grid," and the smaller the adjustment ratio; the larger the dissipation, the greater the discontinuity of the water depth flow field near the "reconstructed correction grid," and the larger the adjustment ratio needs to be.

[0318] Step 8: Topographic water level reconstruction based on adaptive gradient: Using the gradient adjustment ratio generated in step 7, the gradient in the topographic water level gradient method is adjusted to complete the topographic water level reconstruction based on adaptive gradient.

[0319] In step 8, using the gradient adjustment ratio generated in step 7 and the calculation results from the topographic and water level gradient method, the topographic and water level reconstruction gradients of the "reconstruction correction grid" with the larger absolute value are linearly approximated towards the smaller absolute value according to the gradient adjustment ratio. The larger the adjustment ratio, the higher the approximation degree. After the topographic and water level reconstruction gradient is corrected twice, the topography and water depth at each point in the calculation dimension of the "reconstruction correction grid" are recalculated using the relationship between topographic water level and gradient, thus completing the adaptive gradient topographic and water level reconstruction.

[0320] The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber in this invention has the following beneficial effects:

[0321] This invention is based on the topographic water level gradient reconstruction method and flow field change information near complex terrain. By calculating the corresponding equivalent dimensionless wavenumber, an adaptive adjustment ratio is determined to correct the gradient used to reconstruct the topographic water level, thereby completing the adaptive reconstruction of the topographic water level. It can be used to solve the numerical instability problem exhibited by the topographic water level gradient reconstruction method when dealing with complex terrain. It also has the advantages of strict mass conservation, harmony, positive preservation and robustness, and has important academic significance and application value.

[0322] To better understand this invention, specific embodiments have been described in detail above, but these are not intended to limit the invention. Any simple modifications made to the above embodiments based on the technical essence of this invention still fall within the scope of this invention. Each embodiment in this specification focuses on its differences from other embodiments; similar or identical parts between embodiments can be referred to mutually. For system embodiments, since they basically correspond to method embodiments, the description is relatively simple; relevant parts can be referred to the descriptions of the method embodiments.

Claims

1. A second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber, comprising acquiring one-dimensional and two-dimensional hydrodynamic models, characterized in that, It also includes the following steps: Step 1: Classify the dry / wet state of the central grid based on its own dry / wet state and that of its critical grids; The classification of dry and wet types includes: set up Given a one-dimensional region, discretize the region as Ω=U i∈I C i , among which, U i∈I For global grid, grid and in the set of integers As an index set For grid The boundary is calculated on the positive backward side of the dimension. For grid The boundary on the positive front side of the calculated dimension; Unify all grids, so that ∆x is the grid side length; Let b i =b i (t) and η i =η i Let (t) represent the average topography and water level within the i-th grid at time t, then h i =h i (t)=η i (t)-b i (t) represents the water depth corresponding to the i-th grid, b i η i and h i All values ​​are assigned to the center of the grid. Based on the wet / dry state of the central grid, the grid type can be divided into wet grid I. wet (t) and dry grid I dry (t) Two types; Based on the wet / dry state of the central grid's critical grid, the central grid's critical grid is divided into dry grids located at the wet / dry boundary. Wet grid at the wet-dry boundary Wet mesh inside a moist environment and the dry grid inside the dry area Four types; Step 2: For the central grids of different dry and wet types, calculate the reconstruction gradient based on the topographic and water level information of the grids themselves and the critical grids and a specific slope limiter, and perform preliminary reconstruction of the topographic and water level of the central grids; Step 3: For the central grid that is a wet grid or a dry grid located at the wet-dry boundary, the topographic water level of the preliminary reconstruction is corrected according to the topographic water level gradient method; Step 4: Using a second-order precision discontinuity detector, detect the smoothness of the water depth change near the center and critical grid to determine whether the center and critical grid are in the discontinuity zone. Step 5: If the central or critical grid is in a discontinuity region, calculate the equivalent dimensionless wavenumber of the central grid on both the front and back interfaces in the positive direction of the computation dimension; Step 6: Based on the equivalent dimensionless wavenumber, the adaptive dissipation of the central grid is calculated using the relationship between dissipation and equivalent dimensionless wavenumber. Step 7: Normalize the adaptive dissipation to obtain the adjustment ratio θ of the terrain water level gradient; Step 8: Using the adjustment ratio θ, adjust the gradient in the topographic water level gradient method to complete the adaptive gradient topographic water level reconstruction.

2. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 1, characterized in that, The preliminary reconstruction includes: For variables q∈(η,b), the central grid C i The initial linear reconstruction relation within is: ， in, It is the central grid C i The reconstructed gradient of variable q, where x is the coordinate of each point in the positive direction of the computation dimension within the grid, and q i (t) represents the value of the grid center reconstruction variable, x i The coordinates of the center point inside the grid; If the central grid belongs to Configurable settings: ， If the central grid belongs to Or I wet (t), using the minmod slope limiter to compute the gradient for the initial reconstruction of the variable q∈(η,b): ， The minmod function is: ， Where θ typically takes values ​​in the range [1.0, 2.0], q i-1 Reconstruct variable values ​​for the grid behind the central grid, q i Reconstruct variable values ​​for the central grid. Let q be the grid side length. i+1 Here, a1 represents the reconstructed variable value of the grid in front of the central grid, and a1 is the first slope estimate to be compared. m Let m be the m-th slope estimate to be compared, where m is the number of slope estimates to be compared.

3. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 2, characterized in that, A one-sided limit is introduced at the boundary of the central grid, in the central grid C. i boundary At this point, the left limit is: ， The right-hand limit is: 。 4. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 3, characterized in that, Step 3 includes reconstruction correction based on the topographic water level gradient method, including: (1) When it appears hour: ， For variables q∈(η,b), the gradient used to reconstruct the topographic water level is adjusted, denoted as: ， (2) When it appears hour: ， For variables The gradient used to reconstruct the topographic water level is adjusted, denoted as: ， Based on the readjusted gradient For the central grid C i The topography and water levels were reconstructed and corrected: ， in, The water level is on the right side of the rear boundary of the central grid. The terrain to the right of the rear boundary of the central grid. Let t be the water level on the right side of the rear boundary of the central grid. Let t be the water level at the center grid. The terrain to the right of the rear boundary of the central grid at time t; The water level is on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid. Let t be the water level on the left side of the front boundary of the central grid. The terrain to the left of the front boundary of the central grid at time t; Correction values ​​for the topographic water level reconstruction gradient.

5. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 4, characterized in that, When it appears hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections; When it appears or or hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient Make corrections; When it appears hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections; When it appears hour, For the preliminary reconstructed terrain gradient Make corrections, and do not modify the initially reconstructed water level gradient. Make corrections; When it appears hour, For the preliminary reconstructed terrain gradient and the preliminary reconstructed water level gradient All have been revised; When it appears or hour, For the preliminary reconstructed terrain gradient and the initially reconstructed water level gradient All have been revised; When it appears hour, For the preliminary reconstruction of the water level gradient Make corrections, without altering the initially reconstructed terrain gradient. Make corrections; in, This is the reconstruction correction value inside the rear boundary of the central mesh. This is the reconstruction correction value inside the front boundary of the central grid.

6. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 5, characterized in that, The operation method of the intermittent detector includes: Step 41: For water depth information h i h i+2 h i+3 h i+1 h i-1 and h i-2 After normalization, the formula is: denom = max(1.0, max(h) i ,h i+2 ,h i+3 ,h i+1 ,h i-1 ,h i-2 )), Normalization of water depth information: h k1 =h k1 / day; Step 42: Calculate the detection center grid C of the intermittent detector. i discontinuity ψ i The formula is ， a=|h i -h i-1 |+|h i -2h i-1 +h i-2 | a=|h i -h i+1 |+|h i -2h i+1 +h i+2 | ， Step 43: When ψ i ≥ψ c If the water depth change near the detection grid is smooth, the detection grid is considered to be in a discontinuous zone. Among them, k1 = {i-2, i-1, i, i+1, i+2, i+3}, ξ 2 =10 2 ψ c Highly constant.

7. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 6, characterized in that, Step 5 includes the following sub-steps: Step 51: When the center grid C i To reconstruct and correct the mesh and min(ψ) i-1 ,ψ i ,ψ i+1 )≥ψ c At that time, the water depth information h i+1 h i+2 h i+3 h i-2 h i-1 and h i Normalization is performed: denom=max(1.0,max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i )) , h max =max(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / denom, h min =min(h i+1 ,h i+2 ,h i+3 ,h i-2 ,h i-1 ,h i ) / day, Normalization of water depth information: h k2 =h k2 / day, Step 52: Calculate the dimension of the central network forward and backward on both sides of the interface. and Calculate the equivalent dimensionless wavenumber and The formula is ， ， ， ， ， ， ， ， ， ， ， Among them, k2={i+3,i+2,i+1,i-2,i-1,i}.

8. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 7, characterized in that, Step 6 includes determining the equivalent dimensionless wavenumber. and The maximum value in Is it higher than the critical wavenumber k? c , If k ESW Below k c If the reconstructed correction mesh is in a smooth region, then the dissipation γ of the reconstructed correction mesh is considered to be within the smooth region. diss Set to 0.001; If k ESW Higher than k c Then the dissipation γ is used. diss With equivalent dimensionless wavenumber k ESW The dissipation of the reconstructed and corrected mesh is calculated using a specific relation, as follows: ， Where, k max =π.

9. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 8, characterized in that, The adjustment ratio θ of the topographic water level gradient and the dissipation γ diss The relationship is as follows: 。 10. The second-order precision gradient adaptive topographic water level reconstruction method based on dimensionless wavenumber as described in claim 9, characterized in that, The adaptive gradient topography-water level reconstruction includes the following cases: Scenario 1: When hour, and , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Scenario 2: When hour, , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Scenario 3: When hour, , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Scenario 4: When hour, , , For positive, Negative, when hour, , , when hour; , , Scenario 5: When hour, , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Keeping the topography and water level unchanged, the adaptive reconstruction gradients are as follows: ， ， Scenario 6: When hour, , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Scenario 7: When hour, , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Situation 8: When hour, , , Terrain reconstruction gradient Adaptive adjustment and water level reconstruction gradient Keeping constant, the adaptive reconstruction gradients for water level and terrain are as follows: ， ， Situation 9: When hour, , , Negative, For positive, when hour ， ， when hour: ， ， Case 10: When hour, , , Reconstructing the gradient of water level Adaptive adjustment and terrain reconstruction gradient Remaining constant, the adaptive reconstruction gradients for topography and water level are as follows: ， 。

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