Shallow water flow simulation method and system based on source item weak solution LTS
By using a large time-step scheme based on the source term weak solution LTS, the problems of low computational efficiency and spurious flow in shallow water flow simulation are solved, and efficient and accurate shallow water flow simulation is achieved.
Patent Information
- Application Number
- CN202511081635.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-04
- Publication Date
- 2025-11-18
AI Technical Summary
Numerical simulation of shallow water equations in the present technology is inefficient, and traditional methods are prone to spurious flows when dealing with source terms.
A large time step scheme based on source term weak solution LTS is adopted. By dividing the channel into cells and calculating wave velocity and wave intensity at the interface, the state within the cell is updated using a random selection method to avoid sparse wave breakage and overcome the CFL condition limitation.
It improves computational efficiency, avoids spurious flows, and achieves higher computational accuracy and result convergence.
Smart Images
Figure CN120974973A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of shallow water dynamics technology, specifically relating to a shallow water flow simulation method and system based on source term weak solution LTS with high computational efficiency and good accuracy. Background Technology
[0002] Shallow water flow refers to flow where the horizontal scale of motion is much larger than the vertical scale of motion. Its characteristics include negligible vertical velocity and acceleration, resulting in near-static pressure distribution, a free surface, and gravity as the primary driving force. Water flow in rivers can generally be considered shallow water flow.
[0003] Shallow water equations are important mathematical models in hydraulics for describing shallow water flow. Therefore, the simulation of shallow water flow is mainly achieved by solving shallow water equations.
[0004] Shallow water equations are partial differential equations, which are difficult to solve directly. Current numerical simulations of such equations first obtain the Riemann solution at the interface between adjacent elements after discretization, and then update the two elements on either side of the interface. However, this method is computationally inefficient due to limitations imposed by the CFL condition.
[0005] To improve computational efficiency, existing technologies have proposed using a large time-step scheme to simulate nonlinear hyperbolic partial differential equations. By changing the original update method, the limitation of the number of CFLs is overcome, resulting in a significant improvement in computational efficiency.
[0006] The convection part of the shallow water equation is a nonlinear hyperbolic partial differential equation. In addition to the convection part, the bottom slope source term must also be considered. The traditional method is to solve the convection part and then solve the source term as an ordinary differential equation. However, applying this method to the shallow water equation can lead to spurious flow. Summary of the Invention
[0007] Purpose of the invention: To address the above-mentioned shortcomings, this invention provides a shallow water flow simulation method and system based on source term weak solution LTS, which has high computational efficiency, good accuracy, and good result convergence.
[0008] Technical solution: To solve the above problems, the present invention adopts...
[0009] A shallow water flow simulation method based on source term weak solution LTS includes the following steps:
[0010] Initial parameters of the river channel are obtained by dividing the river channel into multiple units along its length and using the initial state of the river flow as the state value within each river channel unit.
[0011] Based on the state value within each river unit, the state between river units is calculated. Based on the state between river units, the state value within each river unit is updated to obtain the shallow water flow state of each river unit at the next moment, thereby realizing the simulation of shallow water flow in the river.
[0012] The inter-channel unit state includes the wave velocity and wave intensity of each wave at the inter-channel unit interface. The source term weak solution is solved to obtain the wave intensity weighting coefficient. The source term weak solution is decomposed into the wave intensity, and the original wave intensity is multiplied by the wave intensity weighting coefficient to obtain the weighted wave intensity. The specific steps for updating the state value within each channel unit include:
[0013] Multiply the wave velocity of each wave calculated at the interface between channel units by the time step, divide by the length of the channel unit, and obtain the value a, the integer part of which is b.
[0014] For a complete wave affecting b channel units, when the wave velocity at the interface between channel units is greater than 0, it indicates that the wave is propagating to the right, and a weighted wave intensity of the wave is added to the b channel units on the right side of the interface between channel units; when the wave velocity at the interface between channel units is less than 0, it indicates that the wave is propagating to the left, and a weighted wave intensity of the wave is added to the b channel units on the left side of the interface between channel units.
[0015] For the (b+1)th channel unit partially affected by the wave, a random number is set. If |ab| is greater than or equal to this random number, then the (b+1)th channel unit is added with a weighted wave strength of the wave.
[0016] This invention also employs a shallow water flow simulation system based on source term weak solution LTS, comprising:
[0017] The parameter determination module is used to obtain the initial parameters of the river channel and divide the river channel into multiple units along its length; the initial value of the river flow is used as the state value within each river channel unit.
[0018] The calculation module is used to calculate the inter-channel state based on the state values within each channel unit; the inter-channel state includes the wave velocity and wave intensity at the interface between channel units, and the weak solution of the source term is decomposed into the wave intensity.
[0019] The update module is used to update the state value within each channel unit according to the state between channel units, so as to obtain the shallow water flow state of each channel unit at the next moment and realize the simulation of shallow water flow in the channel.
[0020] The inter-channel unit state includes the wave velocity and wave intensity of each wave at the inter-channel unit interface. The source term weak solution is solved to obtain the wave intensity weighting coefficient. The source term weak solution is decomposed into the wave intensity, and the original wave intensity is multiplied by the wave intensity weighting coefficient to obtain the weighted wave intensity. The specific steps for updating the state value within each channel unit include:
[0021] Multiply the wave velocity of each wave calculated at the interface between channel units by the time step, divide by the length of the channel unit, and obtain the value a, the integer part of which is b.
[0022] For a complete wave affecting b channel units, when the wave velocity at the interface between channel units is greater than 0, it indicates that the wave is propagating to the right, and a weighted wave intensity of the wave is added to the b channel units on the right side of the interface between channel units; when the wave velocity at the interface between channel units is less than 0, it indicates that the wave is propagating to the left, and a weighted wave intensity of the wave is added to the b channel units on the left side of the interface between channel units.
[0023] For the (b+1)th channel unit partially affected by the wave, a random number is set. If |ab| is greater than or equal to this random number, then the (b+1)th channel unit is added with a weighted wave strength of the wave.
[0024] The present invention also employs a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the above method.
[0025] The present invention also employs a computer-readable storage medium having a computer program stored thereon, characterized in that the computer program, when executed by a processor, implements the steps of the above method.
[0026] Beneficial effects: Compared with the prior art, the significant advantage of this invention is that it uses a large time step scheme of weak solution of bottom slope source term to solve shallow water equations, which improves efficiency and avoids the problem of spurious flow.
[0027] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. Attached Figure Description
[0028] Figure 1 This is the main flowchart of the shallow water flow simulation method based on the large time step scheme of weak source term solution of the present invention.
[0029] Figure 2 This is a schematic diagram of the initial water level and riverbed elevation in an example of the present invention.
[0030] Figure 3 for Figure 1 A flowchart of the shallow water flow simulation steps in the middle channel unit.
[0031] Figure 4 for Figure 3 A flowchart of the steps for calculating the state of the middle unit interface.
[0032] Figure 5 for Figure 3 A flowchart of the state update steps within a unit.
[0033] Figure 6This is a schematic diagram illustrating the principle of a single wave passing through multiple riverbed units in this invention.
[0034] Figure 7 This is a schematic diagram comparing the simulation results calculated according to the method of this invention with the actual solution of shallow water flow in the river in an example of this invention; wherein, Figure 7 (a) in the diagram is a schematic of the true solution. Figure 7 (b) in the figure is a schematic diagram comparing the simulation results of this method with the actual solution. Detailed Implementation
[0035] like Figure 1 As shown, this embodiment presents a shallow water flow simulation method based on source term weak solution LTS (Large Time Step Scheme), which includes the following steps:
[0036] (10) Acquisition of initial parameters of the river channel: Acquire the river channel parameters, including the length, width, bottom elevation of the river channel, and the initial state of the river flow, including the water depth, flow velocity, upstream inlet flow and downstream water level at the initial moment;
[0037] In this embodiment, a channel of uniform width and length of 25m is selected, and distributed along the x-direction (i.e., the length direction of the channel) as follows. Figure 2 As shown, its functional expression satisfies the following relationship:
[0038]
[0039] Among them, z b denoted by , h represents the bottom elevation, and represents the water depth. Initially, the water level in the first half is 5 meters, the bottom elevation is 0, and the water depth is also 5 meters. In the second half, the water level is 2 meters, the bottom elevation is 1 meter, and the water depth is 1 meter. The flow velocity is zero in all sections.
[0040] (20) River channel unit division: The river channel is divided into multiple units along its length, and the initial state of the river flow is used as the state value in each river channel unit. The total length of the river channel is divided by the number of river channel units to obtain the length of each river channel unit.
[0041] In this embodiment, the river channel is divided into 250 units, each unit being 0.1m in length. The flow velocities u1, u2, ..., u in each unit are calculated. 250 and water depths h1, h2, ..., h 250 From formula (1), it can be seen that when the channel element i is less than or equal to 125:
[0042]
[0043] When i is greater than 125:
[0044]
[0045] (30) Shallow water flow simulation of river unit: Based on the state within the river unit, calculate the state between units, update the state within the river unit based on the state between units, and obtain the shallow water flow state of each river unit at the next moment.
[0046] like Figure 3 As shown, the steps for simulating shallow water flow in a river unit include:
[0047] (31) Calculation of interface state between units: Based on the state within the channel unit, the average Roe value of every two adjacent channel units is characterized to obtain the wave velocity and wave intensity at the interface between channel units, and the time step is determined based on the CFL value, such as... Figure 4 As shown;
[0048] (311) Perform Roe averaging on the interface between each segment, for example Figure 6 In In the interface, the (i-1)th unit is on the left and the ith unit is on the right, denoted by L and R respectively. Then for... interface:
[0049]
[0050] for Shallow water wave velocity at the interface This represents the average wave velocity of Roe.
[0051] (312) Calculate the wave velocity and wave intensity. Two waves are emitted from the interface, with the following wave velocities:
[0052]
[0053] Wave strength is:
[0054]
[0055] in:
[0056]
[0057] (313) Decompose the weak solution of the source term into the wave intensity.
[0058] δz=z bR -z bL (11)
[0059] Where z bR Let z be the riverbed elevation of the right unit. bL The elevation of the riverbed in the left unit is shown.
[0060]
[0061]
[0062] Where d L The water surface elevation of the left unit is d. R The elevation of the water surface in the right unit is shown.
[0063]
[0064] Wave strength updated to:
[0065]
[0066] (314) Split a single wave
[0067] To avoid entropy violations in the rarefaction wave breakdown, the original single wave on each side needs to be split into two. Let's assume:
[0068]
[0069] The first wave is:
[0070]
[0071] Corresponding wave intensity is
[0072]
[0073] The second wave is:
[0074]
[0075]
[0076] The final expression for the corresponding wave intensity is:
[0077]
[0078] (32) Intra-unit state update: Based on the wave velocity and wave intensity at the interface between units, the intra-unit state is updated using a random selection method to obtain the shallow water flow state at the next moment.
[0079] like Figure 5 As shown, the state update steps within a cell include:
[0080] (321) Calculate the number of affected elements: Multiply the wave velocity at the interface between elements by the time step, divide by the element length to get the value a, the integer part of which is b;
[0081] for example Figure 6 In As shown in formula (6), this interface will emit two waves with different velocities. After being split in step (314), each interface will emit four waves. Taking one wave velocity as λ as an example, let's assume that the calculated velocity is:
[0082]
[0083] Then the value of a is 2.5, and the value of b is 2.
[0084] (322) Complete wave velocity update of the unit: When the wave velocity at the interface between units is greater than 0, it means that the wave is propagating to the right. Add a weighted wave strength of the wave to the b units on the right side of the unit interface. When the wave velocity at the interface between units is less than 0, it means that the wave is propagating to the left. Add a weighted wave strength of the wave to the b units on the left side of the unit interface. Figure 6 In the example, the method for updating the two cells on the right side of the interface (cell i and cell i+1) is as follows:
[0085] U n+1 =U n +ΔU (26)
[0086] In the above formula, ΔU is the expression in formula (24).
[0087] (323) Partially affected unit wave velocity update: For the b+1th partially affected unit, set a random number between 0 and 1. If |ab| is less than this value, the b+1th unit remains unchanged. If |ab| is greater than or equal to this value, the unit is given a weighted wave strength of the wave.
[0088] like Figure 6 In the example shown, the update method for the (b+1)th unit on the right side of the interface, which is the 3rd unit (i+2 unit), is as follows:
[0089] |ab| is set to 0.5. A random number between 0 and 1 is set. If 0.5 is less than this random number, the third unit remains unchanged. If it is greater than or equal to this random number, the entire weighted wave intensity is added.
[0090] (33) Interface traversal check: Determine whether the update unit is the last interface. If not, proceed to step (31). If yes, end this process.
[0091] (40) Time End Judgment: Determine whether the total calculation time has reached the set total time. If it has, end the simulation; otherwise, set the shallow water flow state of the next moment to the shallow water flow state of the previous moment and proceed to (30) Inter-unit State Calculation Step.
[0092] The true solution is as follows Figure 7 As shown in (a), the comparison between the calculated results and the true solution is as follows: Figure 7As shown in (b) of the diagram. The calculation results show that the adoption of the random selection method increases numerical viscosity and suppresses oscillations; at the same time, the use of a large time step format to update multiple units on both sides of the interface simultaneously breaks through the limitations of the CFL condition and greatly improves the computational efficiency.
Claims
1. A shallow water flow simulation method based on source term weak solution LTS, characterized in that, Includes the following steps: Obtain the initial parameters of the river channel; and divide the river channel into multiple units along its length, using the initial state of the river flow as the state value within each river channel unit. Based on the state value within each river unit, the state between river units is calculated. Based on the state between river units, the state value within each river unit is updated to obtain the shallow water flow state of each river unit at the next moment, thereby realizing the simulation of shallow water flow in the river. The inter-channel unit state includes the wave velocity and wave intensity of each wave at the inter-channel unit interface. The source term weak solution is solved to obtain the wave intensity weighting coefficient. The source term weak solution is decomposed into the wave intensity, and the original wave intensity is multiplied by the wave intensity weighting coefficient to obtain the weighted wave intensity. The specific steps for updating the state value within each channel unit include: Multiply the wave velocity of each wave calculated at the interface between channel units by the time step, divide by the length of the channel unit, and obtain the value a, the integer part of which is b. For a complete wave affecting b channel units, when the wave velocity at the interface between channel units is greater than 0, it indicates that the wave is propagating to the right, and a weighted wave intensity of the wave is added to the b channel units on the right side of the interface between channel units; when the wave velocity at the interface between channel units is less than 0, it indicates that the wave is propagating to the left, and a weighted wave intensity of the wave is added to the b channel units on the left side of the interface between channel units. For the (b+1)th channel unit partially affected by the wave, a random number is set. If |ab| is greater than or equal to this random number, then the (b+1)th channel unit is added with a weighted wave strength of the wave.
2. The shallow water flow simulation method according to claim 1, characterized in that, Two waves emanating from the interface between channel units are split into four waves. When updating the state value within each channel unit, update calculations are performed on all four waves.
3. The shallow water flow simulation method according to claim 1, characterized in that, The steps for simulating shallow water flow in the river channel include: Based on the state values within the channel units, the average Roe of every two adjacent channel units is characterized to obtain the wave velocity and wave intensity at the interface between channel units. The time step is determined based on the CFL value. The wave intensity at the interface between channel units is multiplied by a wave intensity weighting coefficient to obtain the weighted wave intensity. The waves at the interface between channel units are then split to obtain the wave velocity and wave intensity of four waves. Based on the wave velocity and wave intensity at the interface between channel units, the state values within the channel units are updated using a random selection method to obtain the shallow water flow state at the next time step. It is determined whether the updated channel unit is the last channel unit. If not, the state values within the channel units are updated again. If so, the simulation ends.
4. The shallow water flow simulation method according to claim 1, characterized in that, The initial parameters of the river channel include the length, width, bottom elevation, and initial state of the river flow. The initial state of the river flow includes the water depth, flow velocity, upstream inlet flow rate, and downstream water level at the initial moment.
5. The shallow water flow simulation method according to claim 1, characterized in that, After obtaining the shallow water flow state value of each channel unit at the next moment, the total duration of the shallow water flow simulation is calculated. If the total duration reaches the set duration, the simulation ends; otherwise, the shallow water flow state of the channel unit at the next moment is set to the shallow water flow state of the previous moment, and the next cycle is performed to repeat the shallow water flow simulation.
6. A shallow water flow simulation system based on source term weak solution LTS, characterized in that, include: The parameter determination module is used to obtain the initial parameters of the river channel and divide the river channel into multiple units along its length. The initial value of the river flow is used as the state value within each river unit; The calculation module is used to calculate the state between river units based on the state value within each river unit; the state between river units includes the wave velocity and wave intensity at the interface between river units. The source term weak solution is decomposed into the wave intensity. The update module is used to update the state value within each river unit based on the state between river units, so as to obtain the shallow water flow state of each river unit at the next moment, thereby realizing the simulation of shallow water flow in the river. The inter-channel unit state includes the wave velocity and wave intensity of each wave at the inter-channel unit interface. The source term weak solution is solved to obtain the wave intensity weighting coefficient. The source term weak solution is decomposed into the wave intensity, and the original wave intensity is multiplied by the wave intensity weighting coefficient to obtain the weighted wave intensity. The specific steps for updating the state value within each channel unit include: Multiply the wave velocity of each wave calculated at the interface between channel units by the time step, divide by the length of the channel unit, and obtain the value a, the integer part of which is b. For a complete wave affecting b channel units, when the wave velocity at the interface between channel units is greater than 0, it indicates that the wave is propagating to the right, and a weighted wave intensity of the wave is added to the b channel units on the right side of the interface between channel units; when the wave velocity at the interface between channel units is less than 0, it indicates that the wave is propagating to the left, and a weighted wave intensity of the wave is added to the b channel units on the left side of the interface between channel units. For the (b+1)th channel unit partially affected by the wave, a random number is set. If |ab| is greater than or equal to this random number, then the (b+1)th channel unit is added with a weighted wave strength of the wave.
7. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 5.
8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1 to 5.