Single-diagonal implicit Runge-Kutta method for simulating second-order differential equation and application
By combining the single-diagonal implicit Longguta method and Newton iteration, the numerical solution instability and computational load problems of nonlinear rigid differential equations are solved, and efficient and accurate numerical calculations are realized to meet high-precision and long-term simulations. need.
Patent Information
- Application Number
- CN202411497469.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-10-25
- Publication Date
- 2025-01-10
AI Technical Summary
The existing single-diagonal implicit Longguta method is susceptible to numerical coupling effects when solving nonlinear rigid differential equations, resulting in unstable numerical solutions and increased computational load, which cannot meet the needs of high-precision and long-term simulation.
Combining the single-diagonal implicit Longguta method and Newton iteration, iteratively solves the second-order differential equation through Jacobian matrix decomposition and explicit partial prediction, and adjusts the step length by utilizing the convergence of Newton iteration and the stability of the implicit method. To meet convergence conditions, improve the stability and efficiency of numerical calculations.
Provide efficient and accurate numerical solutions when processing second-order nonlinear rigid differential equations, simplify the iteration process, save computing resources, improve the calculation efficiency of the overall numerical solutions, and ensure reliable numerical solutions at each time step.
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Figure CN119294123A_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the technical field of simulation algorithms, and more specifically, to a single diagonal implicit Runge-Kutta method, device, electronic device and computer storage medium for simulating second-order differential equations. Background Art
[0002] Second-order differential equations are widely used in science and engineering, such as vibration problems in physics, dynamic simulation of engineering structures, and biomechanical models in biomedicine. Currently, numerical methods for solving second-order nonlinear stiff differential equations include traditional explicit and implicit methods, as well as more complex multi-step methods and Runge-Kutta methods. Explicit methods such as the forward difference method are simple and easy to use, but for equations containing stiff terms, very small time steps are usually required to ensure numerical stability, which leads to a significant increase in computational cost; implicit methods such as the backward Euler method, the Crank-Nicolson method, and the implicit Runge-Kutta method are more stable when dealing with stiff problems, but solving each time step involves solving a nonlinear or linear algebraic equation system, which greatly increases the amount of computation.
[0003] Compared with the fully implicit Runge-Kutta method, the signal diagonally implicit Runge-Kutta (SDIRK) method has attracted widespread attention because of its low computational complexity and its applicability to stiff differential equations. The numerical format coefficient matrix of the SDIRK method is lower triangular, which makes it easier to construct. In particular, because it is an implicit format, by properly selecting weights and coefficients, the stiff equations can be stably solved at a larger time step, thereby significantly improving the efficiency and accuracy of numerical calculations.
[0004] However, despite the excellent performance of the SDIRK method in many cases, there are still some challenges in the existing implementation. For example, when solving nonlinear stiff equations, the existing SDIRK method may be affected by numerical coupling effects, resulting in instability of the numerical solution or increased computational load during the solution process. In addition, for certain specific application scenarios, such as those requiring high precision and long simulation times, the existing methods cannot provide satisfactory numerical solutions. Therefore, it is an urgent problem to propose a method that can improve the stability, accuracy, and efficiency of numerical calculations for specific categories of second-order differential equations. Summary of the invention
[0005] In response to at least one defect or improvement need in the prior art, the present invention provides a single diagonal implicit Runge-Kutta method and its application for simulating second-order differential equations, aiming to more effectively solve specific categories of second-order differential equations and improve the stability, accuracy and efficiency of numerical calculations.
[0006] To achieve the above-mentioned purpose, according to a first aspect of the present invention, a single diagonal implicit Runge Kutta method for simulating a second-order differential equation is provided, comprising: obtaining a second-order differential equation to be solved and converting it into a target nonlinear equation, calculating the Jacobian matrix of the target nonlinear equation, constructing an iteration matrix according to the Jacobian matrix and decomposing it; using the explicit part of the single diagonal implicit Runge Kutta method to predict the solution of the target nonlinear equation at the next time step to obtain a prediction result; according to the decomposition of the iteration matrix, using the prediction result as the starting value of Newton iteration, the iteration process adopts the single diagonal implicit Runge Kutta method, correcting the prediction result to solve the target nonlinear equation, and repeating the iteration steps until a preset convergence condition is met; performing error estimation according to the solution result of the target nonlinear equation, and calculating a new step size according to the norm of the error estimation, if the error meets the preset condition, advancing the time axis forward by one time step, updating the step size and the solution of the target nonlinear equation, otherwise only updating the step size and returning to perform the calculation of the Jacobian matrix; judging whether to end the simulation according to the simulation time, and storing and displaying the corresponding solution result after the simulation ends.
[0007] In one embodiment of the present invention, the step of transforming the target nonlinear equation includes: constructing the system target nonlinear equation, transforming the n-dimensional second-order differential equation with initial value to be solved Convert to ,in, is a dimensional constant matrix; The expression of the target nonlinear equation obtained by transformation is: , ,in Is an unknown quantity.
[0008] In one embodiment of the present invention, the step of predicting the explicit part of the single diagonal implicit Runge Kutta algorithm comprises: using the explicit part of the single diagonal implicit Runge Kutta algorithm to predict the solution of the next time step. The solution As the predicted value based on the solution of the previous time step , and a known step length The calculation result is: ; .
[0009] In one embodiment of the present invention, the iteration step includes: using the prediction result obtained in the previous time step as the starting value of Newton iteration to solve the solution of the target nonlinear equation in the current time step; the Newton iteration formula of the target nonlinear equation is , , ; Each stage consists of an unknown quantity and The linear equations consist of yes dimensional Jacobian matrix, It's about time function.
[0010] In one embodiment of the present invention, after each Newton iteration is completed, it is determined whether the convergence condition is met. If so, the prediction result of the current step is output according to the following update expression: , ,in ; Otherwise, continue to repeat the iterative steps until the preset convergence condition is met.
[0011] In one embodiment of the present invention, the error estimation step includes: calculating the predicted value and correction value The error between them is estimated to be: in, is the element of the main diagonal of the A matrix in the Butcher table of the single diagonal implicit Runge-Kutta method, is the error value; its norm is calculated based on the allowable error and expressed as: ; The new step size selection method is determined according to the norm of the error estimate: , ;in, Indicates the user-defined absolute tolerance and relative tolerance; if , it means that the calculation result meets the given allowable error, and the time step moves forward according to , , Updates in the form , adjust the step size of the next time step and store the old step size and the old stage derivative vector; if , does not meet the given tolerance, recalculates the new step size, and returns to calculate the Jacobian matrix.
[0012] According to the second aspect of the present invention, there is also provided a single diagonal implicit Runge Kutta device for simulating second-order differential equations, comprising: a target nonlinear equation conversion module, used to obtain the simulated second-order differential equation and convert it into a target nonlinear equation, calculate the Jacobian matrix of the target nonlinear equation, construct an iteration matrix based on the Jacobian matrix and decompose it; an explicit part prediction module, used to use the explicit part of the single diagonal implicit Runge Kutta to predict the solution of the target nonlinear equation at the next time step to obtain a prediction result; a Newton iteration module, used to use the prediction result as the starting value of the Newton iteration according to the decomposition of the iteration matrix, and iterate The iteration process adopts the single diagonal implicit Runge-Kutta method to correct the prediction result to solve the target nonlinear equation, and repeat the iterative steps until the preset convergence condition is met; the error estimation module is used to perform error estimation based on the solution result of the target nonlinear equation, and obtain a new step size based on the norm calculation of the error estimate. If the error meets the preset condition, the time axis is advanced by one time step, and the step size and the solution of the target nonlinear equation are updated; otherwise, only the step size is updated and the calculation of the Jacobian matrix is returned to be executed; the simulation judgment module is used to judge whether to end the simulation according to the simulation time, and store and display the corresponding solution result after the simulation is ended.
[0013] According to the third aspect of the present invention, there is also provided an electronic device, which comprises at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the steps of the method described in any one of the above embodiments.
[0014] According to a fourth aspect of the present invention, there is also provided a computer-readable storage medium storing a computer program executable by an access authentication device, wherein when the computer program runs on the access authentication device, the access authentication device executes the steps of the method described in any one of the above embodiments.
[0015] In general, the above technical solutions conceived by the present invention can achieve at least the following beneficial effects compared with the prior art: The iterative process of solving the second-order differential equation of the present invention is realized by combining the single diagonal implicit Runge Kutta method with the Newton iteration, which can provide efficient and accurate numerical solutions when processing the second-order nonlinear stiff differential equation; the method makes full use of the stability of the single diagonal implicit Runge Kutta method and the convergence of the Newton iteration, so that a reliable numerical solution can be obtained in each time step; The prediction results utilize the explicit part of the single diagonal implicit Runge-Kutta method, which is simple to calculate and does not require solving a set of nonlinear equations or performing complex iterations. It also provides a prediction value that is relatively close to the true solution as the starting value of the Newton iteration, which can significantly reduce the number of Newton iterations. This prediction step that combines the stability of the implicit method with the simplicity of the explicit method can not only simplify the iteration process and save computing resources, but also improve the computational efficiency of the overall numerical solution. BRIEF DESCRIPTION OF THE DRAWINGS
[0016] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.
[0017] Figure 1 A flowchart of a single diagonal implicit Runge-Kutta method for simulating second-order differential equations provided in an embodiment of the present application; Figure 2 A schematic diagram of specific execution steps of a single diagonal implicit Runge-Kutta method for simulating second-order differential equations provided in an embodiment of the present application; Figure 3 A schematic diagram of the Butcher matrix of the 5th-order 4th-level single diagonal implicit Runge-Kutta method provided in an embodiment of the present application; Figure 4 A schematic diagram of the results of solving the van der pol equation using SDIRK4 provided in an embodiment of the present application; Figure 5 A schematic diagram of the structure of a single diagonal implicit Runge-Kutta device for simulating a second-order differential equation provided in an embodiment of the present application; Figure 6 A schematic diagram of the structure of an electronic device provided in an embodiment of the present application; Figure 7 A schematic diagram of the structure of a computer-readable storage medium provided in an embodiment of the present application. DETAILED DESCRIPTION
[0018] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0019] The terms "first", "second", "third", etc. in the specification and claims of this application and the above-mentioned drawings are used to distinguish different objects, rather than to describe a specific order. In addition, the terms "including" and "having" and any variations thereof are intended to cover non-exclusive inclusions. For example, a process, method, system, product or device that includes a series of steps or units is not limited to the listed steps or units, but optionally includes steps or units that are not listed, or optionally includes other steps or units inherent to these processes, methods, products or devices.
[0020] like Figure 1 As shown, the first embodiment of the present invention proposes a teaching engagement evaluation method based on implicit parameter modeling, for example, including: step S1, obtaining a simulated second-order differential equation and converting it into a target nonlinear equation, calculating the Jacobian matrix of the target nonlinear equation, constructing an iteration matrix according to the Jacobian matrix and decomposing it; step S2, using the explicit part of a single diagonal implicit Runge-Kutta method to predict the solution of the target nonlinear equation at the next time step to obtain a prediction result; step S3, according to the decomposition of the iteration matrix, using the prediction result as the starting value of the Newton iteration, and the iteration process adopts a single diagonal implicit Runge-Kutta method. Angle implicit Runge-Kutta method is used to correct the prediction result to solve the target nonlinear equation, and the iterative steps are repeated until the preset convergence condition is met; step S4, error estimation is performed according to the solution result of the target nonlinear equation, and a new step size is calculated according to the norm of the error estimate. If the error meets the preset condition, the time axis is advanced by one time step, and the step size and the solution of the target nonlinear equation are updated; otherwise, only the step size is updated and the calculation of the Jacobian matrix is returned to be executed; step S5, whether to end the simulation is determined according to the simulation time, and the corresponding solution result is stored and displayed after the simulation is ended.
[0021] Specifically, the expression of the single diagonal implicit Runge-Kutta method described in this embodiment is described as follows:
[0022]
[0023] In the formula Represents the intermediate calculation variable of a single-step method.
[0024] The coefficients of the iterative formula can be expressed using Butcher as follows:
[0025] in, The matrix is a lower triangular matrix, and each stage can be calculated separately without generating a large matrix. It has the advantages of a multi-step method. In this case, the algorithm is called the diagonally implicit Runge-Kutta formula (DIRK) algorithm. If the matrix satisfies the same main diagonal elements at the same time, then the solution matrix can be made unchanged. In this case, the algorithm is called the single diagonal implicit Runge-Kutta method. The Butcher matrix expression is as follows:
[0026] In addition, the coefficients satisfy the following conditions: 1)
[0027] 2)
[0028] This condition ensures that the order of the algorithm does not decrease when solving rigid problems.
[0029] Combination Figure 1 As shown, in step S1, firstly, the system target nonlinear equation is constructed according to the second-order differential equation to be solved, and the n-dimensional second-order implicit differential equation with initial value to be solved is converted into Transform expressions of the form:
[0030] in, is a dimensional constant matrix.
[0031] The target nonlinear equation is obtained by using the s-order single diagonal implicit Runge-Kutta method to solve the n-dimensional second-order implicit differential equation:
[0032]
[0033] This process involves solving dimensional nonlinear system of equations, the unknowns are .
[0034] In step S2, the prediction step uses the explicit part of the SDIRK method to predict the solution for the next time step. This prediction is obtained by the previous solution , and a known step length The specific expression is as follows:
[0035]
[0036] This prediction method combines the stability of implicit methods and the simplicity of explicit methods. It can handle stiff problems and reduce computational complexity. It is a better choice for dealing with general and stiff differential equations.
[0037] In step S3, the correction step uses the improved Newton iteration method to correct the predicted value. Specifically, the predicted value obtained in the previous step is used as the starting value of the Newton iteration to more accurately and quickly solve the solution of the current time step.
[0038] The improved Newton iteration formula of the target nonlinear equation is:
[0039]
[0040]
[0041] Each stage of the method consists of an unknown and the system matrix Composition, of which , Both dimensional Jacobian matrix, It's about time Because the other variables are known, we only need to find , we can explicitly solve the unknowns in the target nonlinear equation and In particular, in the single diagonal implicit Runge-Kutta method, we have , so each step only needs to decompose the system matrix once After each step of solving is completed, it is necessary to determine whether the convergence requirements are met. If so, the result of the current step is output according to the following update expression;
[0042]
[0043] Otherwise, continue to repeat the iteration steps until the preset convergence condition is met.
[0044] In step S4, the error estimate needs to be judged after each iteration converges, and the error expression is calculated using the embedded method as follows:
[0045] in, are the elements of the main diagonal of the A matrix in the butcher table, The error value can be Figure 3 Found in.
[0046] The norm is calculated based on the allowable error:
[0047] in, , Indicates the user-defined absolute tolerance and relative tolerance. , indicating that the current error does not meet the allowable error requirements, this step of iteration result cannot be accepted (rejection step), and the step size needs to be adjusted and recalculated; if , indicating that the current error meets the allowable error requirement, the result of this iteration can be accepted (acceptance step), and the time step moves forward.
[0048] The new step size selection method is determined based on the norm of the error estimate, and the standard step size prediction is:
[0049] in is the safety factor, whose value depends on the number of Newton iterations at the current step and the maximum number of Newton iterations , expressed as: . As the prediction of the next step length after the first receiving step, the step length prediction is faster using the above standard step length prediction formula. For non-first receiving steps, the Gustafsson strategy needs to be used to predict the step length:
[0050] Gustafsson's step length prediction formula will be more accurate and safer.
[0051] Taking the Van der pol equation as an example, its expression is:
[0052] The equation is a second-order nonlinear ordinary differential equation (ODE), is a constant, This leads to nonlinear behavior, especially It is obvious when far away from 0. When it is large, it behaves as rigid. Given an initial value , integration interval .
[0053] Figure 4 Three graphs are shown, the first of which gives The second figure shows the function of SDIRK step size and time, where the black solid circles represent acceptable steps and the red "×" represents rejected steps. It can be observed from the figure that very small step sizes are used in steep parts, while larger step sizes can be used for fast calculations in stable parts. The third figure gives the number of Newton iterations required to solve the nonlinear system.
[0054] In summary, the first embodiment of the present invention proposes a single diagonal implicit Runge Kutta method for simulating second-order differential equations. The iterative process of solving the second-order implicit differential equation is achieved by combining the single diagonal implicit Runge Kutta method with the Newton iteration, which can provide efficient and accurate numerical solutions when processing rigid second-order differential equations; the method makes full use of the stability of the single diagonal implicit Runge Kutta method and the convergence of the Newton iteration, so that a reliable numerical solution can be obtained in each time step; the prediction result uses the explicit part of the single diagonal implicit Runge Kutta method, which is simple to calculate and does not require solving a nonlinear system of equations or performing complex iterations, and provides a prediction value that is closer to the true solution as the starting value of the Newton iteration, which can significantly reduce the number of iterations of the Newton iteration. This prediction step that combines the stability of the implicit method and the simplicity of the explicit method can not only simplify the iterative process and save computing resources, but also improve the computational efficiency of the overall numerical solution.
[0055] In addition, if Figure 5 As shown, the second embodiment of the present invention proposes a single diagonal implicit Runge-Kutta device 20 for simulating second-order differential equations, for example, including: a target nonlinear equation conversion module 201, an explicit partial prediction module 202, a Newton iteration module 203, an error estimation module 204 and a simulation judgment module 205.
[0056] Among them, the target nonlinear equation conversion module 201 is used to obtain the simulated second-order differential equation and convert it into the target nonlinear equation, calculate the Jacobian matrix of the target nonlinear equation, construct an iteration matrix according to the Jacobian matrix and decompose it. The explicit part prediction module 202 is used to use the explicit part of the single diagonal implicit Runge Kutta method to predict the solution of the next time step of the target nonlinear equation to obtain a prediction result. The Newton iteration module 203 is used to use the prediction result as the starting value of the Newton iteration according to the decomposition of the iteration matrix. The iteration process adopts the single diagonal implicit Runge Kutta method to correct the prediction result to solve the target nonlinear equation, and repeat the iteration steps until the preset convergence condition is met. The error estimation module 204 is used to perform error estimation according to the solution result of the target nonlinear equation, and calculate a new step size according to the norm of the error estimation. If the error meets the preset condition, the time axis is advanced by one time step, and the step size and the solution of the target nonlinear equation are updated. Otherwise, only the step size is updated and the calculation of the Jacobian matrix is returned. The simulation judgment module 205 is used to judge whether to end the simulation according to the simulation time, and store and display the corresponding solution result after the simulation is ended.
[0057] It is worth mentioning that the method implemented by the single diagonal implicit Runge Kutta device 20 for simulating second-order differential equations disclosed in the second embodiment of the present invention is as described in the first embodiment, so it will not be described in detail here. Optionally, the various modules and the above-mentioned other operations or functions in the second embodiment are respectively for implementing the single diagonal implicit Runge Kutta method for simulating second-order differential equations described in the first embodiment, and the beneficial effects of this embodiment are the same as those of the first embodiment, and for the sake of brevity, they are not repeated here.
[0058] like Figure 6 As shown, the third embodiment of the present invention further proposes an electronic device 30, for example, including: at least one processing unit 31, and at least one storage unit 32, wherein the storage unit 32 stores a computer program, and when the computer program is executed by the processing unit 31, the processing unit 31 executes the method described in the first embodiment, and the beneficial effects of the electronic device 30 provided by this embodiment are the same as the beneficial effects of the single diagonal implicit Runge-Kutta method for simulating second-order differential equations provided by the first embodiment.
[0059] like Figure 7 As shown, the fourth embodiment of the present invention further provides a computer-readable storage medium 40 on which a computer program is stored, and the program implements the steps of the above method when executed by a processor, and the beneficial effects of the computer-readable storage medium 40 provided by this embodiment are the same as the beneficial effects of the single diagonal implicit Runge-Kutta method for simulating second-order differential equations provided by the first embodiment.
[0060] In addition, it can be understood that the aforementioned embodiments are merely illustrative descriptions of the present application. On the premise that the technical features do not conflict, the structures do not contradict, and the purpose of the invention of the present application is not violated, the technical solutions of the various embodiments can be arbitrarily combined and used in combination.
[0061] In the several embodiments provided in the present application, it should be understood that the disclosed systems, devices and / or methods can be implemented in other ways. For example, the device embodiments described above are merely schematic, and the division of the units / modules is merely a logical function division. There may be other division methods in actual implementation, such as multiple units or modules can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the mutual coupling or direct coupling or communication connection shown or discussed can be through some interfaces, indirect coupling or communication connection of devices or units, which can be electrical, mechanical or other forms.
[0062] The units / modules described as separate components may or may not be physically separated, and the components shown as units / modules may or may not be physical units, that is, they may be located in one place or distributed on multiple network units. Some or all of the units / modules may be selected according to actual needs to achieve the purpose of the solution of this embodiment.
[0063] In addition, each functional unit / module in each embodiment of the present application may be integrated into one processing unit / module, or each unit / module may exist physically separately, or two or more units / modules may be integrated into one unit / module. The above-mentioned integrated unit / module may be implemented in the form of hardware or in the form of hardware plus software functional units / modules.
[0064] The above-mentioned integrated unit / module implemented in the form of a software functional unit / module can be stored in a computer-readable storage medium. The above-mentioned software functional unit is stored in a storage medium, including a number of instructions for enabling one or more processors of a computer device (which can be a personal computer, a server, or a network device, etc.) to perform some steps of the method described in each embodiment of the present invention. The aforementioned storage medium includes: U disk, mobile hard disk, read-only memory (Read-Only Memory, referred to as ROM), random access memory (Random Access Memory, referred to as RAM), disk or optical disk and other media that can store program codes.
[0065] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit it. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present application.
Claims
1. A single diagonal implicit Runge-Kutta method for simulating second-order differential equations, characterized in that: include: Acquire a simulated second-order differential equation and convert it into a target nonlinear equation, calculate the Jacobian matrix of the target nonlinear equation, construct an iterative matrix according to the Jacobian matrix and decompose it; Using the explicit part of the single diagonal implicit Runge-Kutta method to predict the solution of the target nonlinear equation at the next time step, a prediction result is obtained; According to the decomposition of the iteration matrix, the prediction result is used as the starting value of Newton iteration, the iteration process adopts the single diagonal implicit Runge-Kutta method, the prediction result is corrected to solve the target nonlinear equation, and the iteration steps are repeated until the preset convergence condition is met; An error is estimated according to the solution of the target nonlinear equation, and a new step size is calculated according to the norm of the error estimate. If the error satisfies a preset condition, the time axis is advanced by one time step, and the step size and the solution of the target nonlinear equation are updated. Otherwise, only the step size is updated and the calculation of the Jacobian matrix is returned to be executed. Determine whether to end the simulation based on the simulation time, and store and display the corresponding solution result after the simulation is ended.
2. The single diagonal implicit Runge-Kutta method for simulating second-order differential equations according to claim 1, characterized in that: The step of transforming the target nonlinear equation comprises: Construct the system target nonlinear equation and convert the n-dimensional second-order differential equation with initial value to be solved Convert to ,in, is a dimensional constant matrix; The expression of the target nonlinear equation obtained by transformation is: , ,in Is an unknown quantity.
3. The single diagonal implicit Runge-Kutta method for simulating second-order differential equations according to claim 2, characterized in that: The steps of the explicit partial prediction of the single diagonal implicit Runge-Kutta method include: Use the explicit part of the single diagonal implicit Runge-Kutta method to predict the solution for the next time step The solution As the predicted value based on the solution of the previous time step , and known step length The calculation result is: ; .
4. The single diagonal implicit Runge-Kutta method for simulating second-order differential equations according to claim 3, characterized in that: The iterative steps include: Using the prediction result obtained in the previous time step as the starting value of Newton iteration, solving the solution of the target nonlinear equation in the current time step; The Newton iteration formula of the target nonlinear equation is: , , ; Each stage consists of an unknown quantity and The linear equations consist of yes dimensional Jacobian matrix, It's about time function.
5. The single diagonal implicit Runge-Kutta method for simulating second-order differential equations according to claim 4, characterized in that: After each Newton iteration is completed, it is determined whether the convergence condition is met. If so, the prediction result of the current time step is output according to the following update expression: , ,in ; Otherwise, continue to repeat the iterative steps until the preset convergence condition is met.
6. The single diagonal implicit Runge-Kutta method for simulating second-order differential equations according to claim 5, characterized in that: The error estimation step comprises: Calculate predicted values and correction value The error between them is estimated to be: in, are the elements of the main diagonal of the A matrix in the Butcher table of the single diagonal implicit Runge-Kutta method, is the error value; The norm is calculated based on the allowable error: ; The new step size selection method is determined according to the norm of the error estimate: , ;in, , Indicates the customized absolute tolerance and relative tolerance; like , it means that the calculation result meets the given allowable error, and the time step moves forward according to , , Updates in the form , adjust the step size of the next time step and store the old step size and the old stage derivative vector; if , does not meet the given tolerance, recalculates the new step size, and returns to calculate the Jacobian matrix.
7. A single diagonal implicit Runge-Kutta device for simulating second-order differential equations, characterized in that: include: A target nonlinear equation conversion module is used to obtain a simulated second-order differential equation and convert it into a target nonlinear equation, calculate the Jacobian matrix of the target nonlinear equation, construct an iterative matrix according to the Jacobian matrix and decompose it; An explicit partial prediction module, used for predicting the solution of the target nonlinear equation at the next time step by using the explicit part of the single diagonal implicit Runge-Kutta method to obtain a prediction result; A Newton iteration module, for taking the prediction result as the starting value of Newton iteration according to the decomposition of the iteration matrix, the iteration process adopts a single diagonal implicit Runge-Kutta method, corrects the prediction result to solve the target nonlinear equation, and repeats the iteration steps until a preset convergence condition is met; An error estimation module is used to perform error estimation according to the solution result of the target nonlinear equation, and calculate a new step size according to the norm of the error estimate. If the error meets the preset condition, the time axis is advanced by one time step, and the step size and the solution of the target nonlinear equation are updated; otherwise, only the step size is updated and the calculation of the Jacobian matrix is returned to be executed; The simulation judgment module is used to judge whether to end the simulation according to the simulation time, and store and display the corresponding solution result after the simulation is ended.
8. An electronic device, characterized in that: The method comprises at least one processing unit and at least one storage unit, wherein the storage unit stores a computer program, and when the computer program is executed by the processing unit, the processing unit executes the steps of the method according to any one of claims 1 to 6.
9. A computer-readable storage medium, characterized in that: It stores a computer program executable by the access authentication device. When the computer program runs on the access authentication device, the access authentication device executes the steps of the method according to any one of claims 1 to 6.