Space-time high-precision structure-preserving numerical algorithm
The original equation is linked with physical properties through the Lagrange multiplier method and transformed into optimization problems, solving the problem of insufficient accuracy in the time direction of traditional algorithms, and realizing a high-precision structure-guaranteed numerical algorithm with high precision in space-time.
Patent Information
- Application Number
- CN202411314695.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-09-20
- Publication Date
- 2025-02-07
AI Technical Summary
Traditional structure-saving numerical algorithms are difficult to achieve high-precision calculations in the time direction, resulting in insufficient calculation accuracy.
The Lagrange multiplier method is used to link the original equation or system of equations with related physical properties, and transform it into nonlinear equation constraint optimization problems with partial differential equation constraints, and solve it through high-precision numerical discrete and optimization methods.
It realizes a high-precision structure-keeping numerical algorithm with high precision in space and time, maintains the physical properties of the original problem, improves the calculation accuracy, and is suitable for long-term numerical calculations.
Smart Images
Figure CN119397140A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of algorithm technology, and in particular to a high-precision structure-preserving numerical algorithm in time and space. Background Art
[0002] Many partial differential equations or systems of equations obtained by modeling physical problems often need to satisfy certain physical properties for their exact solutions, such as the classic nonlinear Schrödinger equation in quantum mechanics that satisfies mass and energy conservation, and the Allen-Cahn and Cahn-Hilliard equations in the gradient flow model that satisfy energy dissipation. When designing numerical algorithms, we often need to maintain the physical properties of the original equation or system of equations as much as possible, that is, preserve the structure, so that the algorithm has better robustness.
[0003] Traditional structure-preserving algorithms start from the perspective of numerical discretization, use finite difference methods, finite element methods or spectral methods to perform numerical discretization, construct corresponding numerical formats and ensure their corresponding physical properties in the discrete case. Such formats are usually called structure-preserving numerical formats. Most numerical formats can achieve high-order accuracy in the spatial direction (such as using high-order finite difference methods or spectral methods), while the numerical accuracy in the time direction is mostly second-order. Traditional structure-preserving algorithms cannot effectively improve the calculation accuracy. Based on this, a high-precision structure-preserving numerical algorithm in space and time is proposed. Summary of the invention
[0004] The purpose of the present invention is to solve the problems existing in the prior art and to propose a high-precision structure-preserving numerical algorithm in time and space.
[0005] In order to achieve the above object, the present invention adopts the following technical solutions:
[0006] A high-precision structure-preserving numerical algorithm in time and space, the algorithm specifically includes the following process:
[0007] S1. Based on the Lagrange multiplier method, the physical quantities of the relevant physical properties in the original equation or equation group are also regarded as variables. By introducing an appropriate number of Lagrange multipliers, the original equation or equation group is combined with the relevant physical properties to establish the entire problem;
[0008] S2. Combining optimization ideas, the entire problem is transformed into a nonlinear equality-constrained optimization problem with partial differential equations or equation groups.
[0009] S3, high-precision numerical discretization of the time and space directions of the nonlinear equality-constrained optimization problem in S2;
[0010] S4. Select an appropriate optimization method to solve the optimization problem in the discrete case, and then the numerical calculation of the entire algorithm can be realized.
[0011] Preferably, in S1, the entire problem is converted into a higher dimensional system with multiple variables, and the solution of the original equation or system of equations is a subset of the solution of this higher dimensional new system. Considering the equivalence of the new system with the original equation or system of equations, the exact solution of the corresponding Lagrange multiplier can be obtained, and then the solution to the entire problem is converted into the solution of the problem of calculating the new system.
[0012] Preferably, the newly established system in S2 is used as a constraint condition of the optimization problem, and the minimum value of the sum of errors between a number of Lagrange multipliers and their corresponding exact solutions is used as the objective function of the optimization problem.
[0013] Preferably, S3 is specifically as follows: discretizing the objective function and the constraints respectively, the discretization of the objective function is based on the composite trapezoidal formula in numerical integration, the discretization of the constraints selects a high-precision numerical method, and an appropriate optimization method is selected to solve the optimization problem in the discrete case, thereby realizing the numerical calculation of the entire algorithm.
[0014] Compared with the prior art, the present invention has the following beneficial effects:
[0015] 1. The algorithm proposed in the present invention naturally adds relevant physical properties to the newly constructed system. The entire optimization problem well guarantees the physical properties of the original problem and provides a platform for the design of high-precision numerical algorithms. The continuous problem naturally maintains the original physical structure, that is, the corresponding high-precision discrete algorithm also naturally guarantees the discrete physical properties.
[0016] 2. Compared with the traditional numerical discretization method, the new algorithm proposed in the present invention is constructed from a new perspective of optimization, and based on the characteristic that higher-dimensional systems naturally guarantee the original physical properties, it has greater freedom in the construction of structure-preserving numerical algorithms.
[0017] 3. The new algorithm proposed in the present invention has the advantage of maintaining the original physical structure, has good algorithm stability when solving related problems and numerical simulations, and is suitable for long-term numerical calculations. BRIEF DESCRIPTION OF THE DRAWINGS
[0018] Figure 1 The figure is a flow chart of a high-precision structure-preserving numerical algorithm in time and space proposed by the present invention. DETAILED DESCRIPTION
[0019] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.
[0020] Example, see Figure 1 , a high-precision structure-preserving numerical algorithm in time and space. The algorithm proposed in this scheme is based on the Lagrange multiplier method. The physical quantities of relevant physical properties in the original equation or equation group, such as mass, energy, etc., are also regarded as variables. By introducing an appropriate number of Lagrange multipliers, the original equation or equation group is combined with the relevant physical properties, thereby converting the entire problem into a higher-dimensional system with multiple variables;
[0021] Based on the above, it can be concluded that the solution of the original equation or system of equations is a subset of the solution of this higher-dimensional new system; considering the equivalence of the new system to the original equation or system of equations, the exact solution of the corresponding Lagrange multiplier can be obtained, and then the solution to the entire problem can be transformed into the solution to the problem of calculating the new system.
[0022] When the algorithm is implemented, the actual solution results of the introduced Lagrange multipliers are taken into consideration and combined with the optimization idea, the entire problem is transformed into a type of nonlinear equality constrained optimization problem with partial differential equations or equation groups constraints, in which the new system established above serves as the constraint condition of the optimization problem, and the minimum value of the sum of errors between several Lagrange multipliers and their corresponding exact solutions serves as the objective function of the optimization problem.
[0023] Since the relevant physical properties have been naturally incorporated into the newly constructed system, the entire optimization problem well guarantees the physical properties of the original problem and provides a platform for the design of high-precision numerical algorithms.
[0024] Specifically, based on the above optimization problem, the objective function and constraint conditions are discretized respectively. The discretization of the objective function is based on the composite trapezoidal formula in numerical integration, and the discretization of the constraint conditions uses a high-precision numerical method;
[0025] Specifically, they include: high-order finite difference methods or spectral methods in the spatial direction; and the algebraically stable Runge-Kutta method in the time direction, such as the diagonal implicit Runge-Kutta method, Gaussian collocation method or high-order backward Euler format, thereby obtaining a class of high-precision numerical algorithms in space and time.
[0026] Since continuous problems naturally maintain the original physical structure, the corresponding high-precision discrete algorithms also naturally guarantee discrete physical properties. Finally, by selecting appropriate optimization methods and solving the optimization problem in the discrete case, the numerical calculation of the entire algorithm can be realized.
[0027] Specifically, the algorithm is explained using the classic nonlinear Schrödinger equation in quantum mechanics as an example:
[0028] Consider the following d (d = 1, 2, 3)-dimensional nonlinear Schrödinger equation:
[0029]
[0030] Under appropriate boundary conditions, it can be proved that the above initial boundary value problem satisfies the following mass conservation and energy conservation:
[0031]
[0032] Since the equation has a variational structure, the original equation can be transformed into the following form using complex variational derivatives:
[0033]
[0034] In the design of the new algorithm, energy and mass are considered equally as variables, and the same number of Lagrange multipliers are introduced.
[0035] In order to ensure the structural stability of the system, the equations of two conservation quantities are also added, and the entire problem is equivalently transformed into a higher-dimensional system with multiple variables:
[0036]
[0037] Here and Usually they are chosen as the variational derivatives of energy and mass with respect to the original function, that is, and The two Lagrange multipliers η(t) and ζ(t) here are used to constrain the conservation of energy and mass.
[0038] Obviously, when both multipliers are 0, the new system degenerates into the original nonlinear Schrödinger equation, that is, the new system has a solution (ψ, η = 1, ζ = 0), where ψ is the solution of the original nonlinear Schrödinger equation. According to the above analysis, when designing numerical algorithms for numerical calculations, we try to find solutions where the two Lagrange multipliers are close to 0. Based on this principle, we introduce the optimization idea and consider the optimization problem with PDE constraints. Assuming t = t n When the numerical solution has been obtained, at t∈(t n , t n+1] The above system is transformed into the following nonlinear equality constrained optimization problem:
[0039]
[0040] Obviously, when both Lagrange multipliers are 0, the optimization system is equivalent to the above high-dimensional system and the original problem, so the subsequent numerical algorithm design will be based on the optimization problem.
[0041] For high-precision discretization of space, Fourier pseudospectrum or sine pseudospectrum is selected for numerical discretization based on the regularity and boundary conditions of the solution. For high-precision discretization of time, the algebraically stable Runge-Kutta method is selected. Since the new system naturally satisfies the conservation of energy and mass, the numerical format after discretization also naturally satisfies the conservation of energy and mass in the discrete case.
[0042] Considering the existence of Lagrange multipliers, it is necessary to solve nonlinear algebraic equations at each level to determine solutions with two Lagrange multipliers close to 0. Considering that the discretization of the Runge-Kutta method in the time direction will produce multiple (greater than or equal to 2) multiplier forms, and there are only two equality constraints, the Lagrange-Newton method is used to calculate the optimization problem, so as to obtain a solution with multipliers close to 0.
[0043] The specific algorithm flow is as follows:
[0044] Step 1: Perform high-precision numerical discretization on the time and space directions of the above optimization problem; Step 2: For the given result (initial value) of the 0th layer, use the first constraint condition to obtain the expression of the 1st layer numerical result with respect to the Lagrange multiplier;
[0045] Step 3: Substitute the expression obtained in the first step into the following two constraints, use the Lagrange-Newton method combined with the objective function to solve the optimization problem, and obtain the value of the Lagrange multiplier;
[0046] Step 4: Substitute the multiplier result back into the expression obtained in the first step to obtain the numerical result of the first layer;
[0047] Step5: Repeat Step1-Step3 to obtain the relevant numerical results that need to be calculated within a limited time.
[0048] Compared with the traditional numerical discretization method, the new algorithm is constructed from a new perspective of optimization, and based on the characteristic that higher-dimensional systems naturally guarantee the original physical properties, it has greater freedom in the construction of structure-preserving numerical algorithms, especially high-precision numerical discretization in time. Although each layer in the entire numerical calculation adds an optimization problem calculation, based on the selection of the optimization method, the increase in the amount of calculation of the entire problem is not very obvious, and the overall calculation efficiency is comparable to that of traditional numerical algorithms.
[0049] Finally, since the new algorithm has the advantage of maintaining the original physical structure, it has good algorithm stability when solving related problems and numerical simulations, and is suitable for long-term numerical calculations.
[0050] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.
Claims
1. A high-precision structure-preserving numerical algorithm in time and space, characterized in that: The algorithm specifically includes the following processes: S1. Based on the Lagrange multiplier method, the physical quantities of the relevant physical properties in the original equation or equation group are also regarded as variables. By introducing an appropriate number of Lagrange multipliers, the original equation or equation group is combined with the relevant physical properties to establish the entire problem; S2. Combining optimization ideas, the entire problem is transformed into a nonlinear equality-constrained optimization problem with partial differential equations or equation groups. S3, high-precision numerical discretization of the time and space directions of the nonlinear equality-constrained optimization problem in S2; S4. Select an appropriate optimization method to solve the optimization problem in the discrete case, and then the numerical calculation of the entire algorithm can be realized.
2. A high-precision structure-preserving numerical algorithm in time and space according to claim 1, characterized in that: In S1, the entire problem is converted into a higher dimensional system with multiple variables. The solution of the original equation or system of equations is a subset of the solution of this higher dimensional new system. Considering the equivalence of the new system with the original equation or system of equations, the exact solution of the corresponding Lagrange multiplier can be obtained, and then the solution to the entire problem is converted into the solution of the problem of calculating the new system.
3. A high-precision structure-preserving numerical algorithm in time and space according to claim 1, characterized in that: In S2, the newly established system is used as a constraint condition of the optimization problem, and the minimum value of the sum of errors between a number of Lagrange multipliers and their corresponding exact solutions is used as the objective function of the optimization problem.
4. A high-precision structure-preserving numerical algorithm in time and space according to claim 1, characterized in that: The S3 is specifically as follows: discretizing the objective function and the constraints respectively. The discretization of the objective function is based on the composite trapezoidal formula in numerical integration. The discretization of the constraints uses a high-precision numerical method. An appropriate optimization method is selected to solve the optimization problem in the discrete case, thereby realizing the numerical calculation of the entire algorithm.