High-precision numerical method for solving Wallasov equation in high-dimensional space by using structured grid

Abstract

The invention discloses a high-precision numerical method for solving a wallasov equation in a high-dimensional space, and belongs to the field of plasma physics. The method mainly overcomes the defects existing in an existing plasma PIC calculation method, and the calculation accuracy and precision are improved. According to the method, a PIC method is replaced by a method for directly solving a Warzisov equation in a high-dimensional space. The method has the advantages that noise generated by a limited number of particles in a PIC method can be avoided, and fine structures in plasma motion can be captured. The maximum difficulty of direct solving is from high dimension, because the increase of the calculated amount is increased according to N2d, N is the size of a grid, and d is the dimension of a physical space. The spectral element method adopted by the invention has the advantages of high-order precision, easiness in parallelization, easiness in interpolation, suitability for long-time integration and the like. The feasibility, the correctness and the superiority of the method are proved by comparing the calculation theory of the six-dimensional space with the result of the PIC method.

Description

A High Order Numerical Method of Using Structured Grid on SolvingVlasov Equation in High Dimensions Technical Field

[0001] The invention belongs to the field of plasma numerical solution, and particularly relates to a technology of a high-precision numerical method for solving the Vlasov equation in high-dimensional space by using a structured grid. Background Art

[0002] Plasma computing refers to the study of the properties and behaviors of plasma through numerical calculations and simulation methods. Plasma is an ionized gaseous substance composed of positive and negative ions produced by the ionization of atoms and atomic groups that have been deprived of some electrons. It is widely present in the universe and is often regarded as the fourth state of matter.

[0003] The Boltzmann equation or Boltzmann transport equation (BTE) is a mathematical model that describes the motion of plasma. It is a partial differential equation that describes the statistical behavior of thermodynamic systems in non-thermodynamic equilibrium states, proposed by Ludwig Boltzmann in 1872. The Boltzmann equation is a nonlinear integro-differential equation. The unknown function in the equation is a six-dimensional probability density function that contains the spatial position and momentum of the particle. The Boltzmann equation can be used to determine how physical quantities change, such as the heat energy and momentum of a fluid during transport. Until 2010, the exact solution of the Boltzmann equation was mathematically proven to be well-behaved. This means that if a perturbation is applied to a system that obeys the Boltzmann equation, the system will eventually return to equilibrium, rather than diverging to infinity or exhibiting other behaviors. However, this existence proof does not help us solve the equation in real problems. In fact, this conclusion only tells us whether solutions exist under certain conditions, but not how to find them. In practice, numerical methods are used to find approximate solutions to various forms of the Boltzmann equation, with applications ranging from hypersonic aerodynamics in rarefied air flows to the flow of plasmas.

[0004] Plasma and charged particle computing are of great significance in scientific and engineering practice. There are three different approaches to plasma computing: microscopic models, kinetic models, and fluid models. Microscopic models are governed by the Liouville equations. For a system with N particles, each charged particle is represented by 6 variables (x, y, z, v x 、v y 、v z) description, with a total of 6N variables. This requires solving the Liouville equations in a 6N-dimensional phase space, which obviously exceeds the capabilities of current supercomputers. On the other hand, the simplest model is the fluid model, which treats the plasma as a conductive fluid with electromagnetic forces acting on it inside, which leads to the need to solve the magnetohydrodynamic (MHD) equations in three dimensions (x, y, and z). MHD solves average quantities, such as density and charge, which makes it unable to describe the fine structure inside the plasma. Due to the limitations of computer speed, MHD is a realistic and feasible method in current plasma calculations. In between these two models is the kinetic model, which solves the six-dimensional (x, y, z, v) x 、v y 、v z ) to solve the charge density function. The Vlasov equation describes the evolution of a particle system under the action of an internal collisionless, self-consistent electromagnetic field. This patent is about the calculation method of the dynamic model.

[0005] There are currently two different approaches to solving the dynamic model. The most popular approach is to use macroscopic particles to represent particle clusters and bunches, and to move the macroscopic particles along the characteristic lines of the Vlasov equation. This is the so-called PIC (Particle-In-Cell) method, which uses the Lagrange-Euler method to solve the Vlasov equation using the law that particles move along the characteristic lines of their image space. In principle, it simplifies partial differential equations (PDEs) into ordinary differential equations (ODEs). The interaction between charged particles, called the space charge effect, is calculated in the following way: first, the charge carried by the particle cluster is distributed to a fixed grid, and then the Poisson equation is solved on the fixed grid to obtain the electrostatic potential component. Finally, the electrostatic field can be calculated by the electrostatic potential step. The PIC method has the following advantages: it is fast and easy to implement, but similar to MHD, it is also difficult to describe the detailed structure in the plasma. In addition, it has the noise phenomenon caused by the use of a limited number of macroparticles. When the number of particles N increases, the size of this noise increases. The speed decreases very slowly. Another way to solve the dynamic model is to solve it directly using the Varasov equation. This can overcome the shortcomings of the PIC method, but due to the high-dimensional nature of the Varasov equation, numerical simulations have only been performed in low dimensions (such as two-dimensional 1P1V or axisymmetric cases). Four-dimensional 2P2V simulations are very rare. Due to the increase in speed and memory of supercomputers, the six-dimensional 3P3V problem will be solvable, and this patented technology is applied in this direction.

[0006] The numerical methods for solving the Vlasov equation can generally be divided into Fourier spectral method, finite element method, finite difference method, and finite volume method. The Fourier spectral method has been used for computational domains with periodic boundary conditions. It can only be applied to conventional computational domains. In order to handle calculations in complex geometric regions, the finite element method is more appropriate. Since the finite element method usually uses a low-order expansion basis, the only way to improve the accuracy is to increase the number of elements in the computational domain. This reduces its computational efficiency. The finite difference method can achieve high-order accuracy by using a high-order format, but it is generally limited to regular computational domains. Similar to the finite element method, the finite difference method can also be applied to irregular domains using the so-called cutting element scheme. In 1976, Cheng and Knorr published their groundbreaking semi-Lagrangian method paper for solving the Vlasov equation on a fixed grid in phase space. Unlike the general method of directly discretizing the phase space to solve the Vlasov equation, it is based on the characteristic line theory of the phase space to solve it, without the need to discretize the phase space, and shows better performance than the general method.

[0007] Most of the above plasma calculations use low-order precision finite element method, finite difference method, finite volume method and semi-Lagrangian method, and many artificial corrections are used to stabilize the calculation. This patent uses high-order precision methods and large-scale calculations to avoid these shortcomings. For high-order methods, this patent uses the spectral element method (Spectral Element Method). It combines the flexibility of the finite element method and the high-order precision of the spectral method. It also has the advantages of easy interpolation and high precision, because the interpolation accuracy can be easily improved by increasing the base order of the polynomial expansion. For large-scale calculations, SEM also has the advantage of easy parallelization. Using SEM, this patent method develops a scalable parallel Vlasov solver, especially for six-dimensional problems, which is an area where there is not much work at present. In addition, most of the existing high-order precision methods for plasma calculations are based on finite volume and finite difference methods, while the high-order precision methods based on the semi-Lagrangian (SLM, Semi-Lagrange Method) method have better performance. Summary of the invention

[0008] In order to overcome the computational difficulties in high-dimensional space, we propose a parallel direct solution to the Vlasov equations based on the spectral element method (SEM). It uses a direct method to replace the usual PIC method for plasma calculations in dynamic space to solve the Vlasov-Poisson equation system. There are several advantages to directly solving the Vlasov equations, such as avoiding the noise generated by the PIC method due to the use of a limited number of particles and capturing the fine structure in the plasma motion. The biggest difficulty in directly solving the Vlasov equations comes from high dimensions, because the amount of calculation increases as N 2d growth, N is the grid size, d is the dimension of the physical space. The spectral element method (SEM) has the advantages of high-order accuracy, easy parallelization, easy implementation of interpolation calculations, and long-term integration due to its high-order accuracy. Parallel decomposition technology is used in high-dimensional calculation areas, including a 3D Poisson solver suitable for parallel expansion. Calculations have been performed on plasma examples of various situations, and they are in good agreement with theoretical and standard results. The effectiveness and superiority of the method are demonstrated.

[0009] The technical solution of the present application is: a high-precision numerical method for solving the Varasov equation in high-dimensional space using a structured grid, by using a high-order precision spectral element method on a structured grid when directly solving the Varasov equation, to accurately calculate the distribution function of the particles and perform interpolation calculations. At the same time, large-scale parallel computing is achieved by simultaneously performing regional decomposition on two orthogonal subspaces, solving the difficulties of high-dimensional computing. This is achieved by decomposing the high-dimensional phase space into two mutually orthogonal subspaces: low-dimensional subspaces of physics and velocity. Then, the time splitting integration format is applied to the subspace, and the corresponding parts of the Varasov equation are discretized in three steps in the physical, velocity and physical subspaces, and the semi-Lagrangian method is used for time integration to complete the time integral solution of the original Varasov equation in high-dimensional space. The electric field intensity at each grid point required for the time integral in the velocity space is obtained by solving the Poisson equation in the physical space.

[0010] Furthermore, the method specifically includes the following steps:

[0011] S1. Analysis of the six-dimensional Vlasov-Poisson equations based on the dimensions and spatial shape of the plasma calculations;

[0012] S2, using a suitable grid to discretize the corresponding subspace;

[0013] S3, selecting a basis function, and expanding the particle probability density distribution field of each volume unit and the electrostatic field and current density field in the physical subspace using the basis function;

[0014] S4, solving Poisson's equation in computational physical space to obtain the electrostatic field at each grid point;

[0015] S5. Combine the conservation conditions to form a time-dependent semi-discrete equation on the two subspaces;

[0016] S6. In terms of time, the implicit semi-Lagrangian (SLM) method is used to complete the time integration in three steps;

[0017] Furthermore, the Vlasov-Poisson equations refer to the Vlasov equations on 3P3V and the Poisson equations on three-dimensional space.

[0018] Furthermore, the discrete solution region in step S2 is specifically: the subspace is a three-dimensional hexahedral region;

[0019] Furthermore, before step S3, it also includes considering regional decomposition to realize parallel calculation. Before the calculation starts, each processor processes and initializes the corresponding physical and velocity subspaces;

[0020] Furthermore, all the adjacent unit interfaces on the same processor and different processors need to meet continuity conditions.

[0021] Furthermore, step S4 specifically comprises: satisfying the conservation condition of the total mass by the physical quantities of all units in all subspaces, thereby forming a semi-discrete equation related to time.

[0022] Furthermore, step S5 is specifically as follows:

[0023] First, the high-dimensional Vlasov-Poisson equations in step S1 are solved by applying the implicit semi-Lagrangian method (SLM);

[0024] Then, the semi-discrete equations obtained in step S4 are combined to obtain discrete equations on two corresponding subspaces.

[0025] Further, step S6 is specifically as follows: firstly, according to the discrete equation on the physical subspace obtained in step S5, a time integration of the half-step physical subspace is completed;

[0026] Perform spatial integration in the velocity subspace to calculate the current intensity and electric field intensity distribution at each physical space grid point;

[0027] Use the electrostatic field to update the velocity field and complete a one-step time integration in the velocity subspace;

[0028] Then return to the physical subspace and use the latest velocity field to complete another half-step time integration in the physical subspace.

[0029] Beneficial effects of the present invention: A high-precision numerical method for solving the Varasov equation in high-dimensional space using a structured grid is provided by the present invention. The high-dimensional space is decomposed into two mutually orthogonal subspaces, and the Varasov equation is projected into the two subspaces for step-by-step integration and solution. The physical quantities in the two subspaces are expanded using the spectral element method, and time integration is completed in three substeps in the physical, velocity and physical subspaces respectively. The electrostatic field in the physical subspace is obtained by solving the Poisson equation. This method can ensure high precision of physical quantity discretization, high precision of interpolation calculation and high precision of numerical integration, thereby ensuring the correct result of long-term integration; at the same time, since parallel computing methods are used in both subspaces, the calculation scale is greatly increased, which can meet the huge calculation amount required for six-dimensional space calculation; in addition, since the method of the present invention uses the SLM method, it has implicit characteristics and satisfies unconditional stability, and due to the good parallel scalability of the spectral element method, the calculation accuracy is greatly improved and the parallel efficiency is good, and a large-scale supercomputer can be used. BRIEF DESCRIPTION OF THE DRAWINGS

[0030] Figure 1 is a flow chart of a high-precision numerical method for solving the Varasov equation in high-dimensional space using a structured grid according to the present invention;

[0031] Figure 2 It is a schematic diagram of the mapping relationship of decomposing the six-dimensional space into two three-dimensional spaces in the present invention;

[0032] Figure 3 Schematic diagram of the hexahedral surface mesh in three-dimensional space used in the present invention (P=4, E=64);

[0033] Figure 4 is a schematic diagram of a mesh on a hexahedral unit in a three-dimensional space in the present invention (P=4);

[0034] Figure 5 Schematic diagram of the expansion base used on the hexahedral unit in the three-dimensional space used in the present invention

[0035] Figure 6 It is a schematic diagram of the parallel data structure in the present invention.

[0036] Figure 7 .Shur matrix decomposition method diagram DETAILED DESCRIPTION

[0037] To facilitate those skilled in the art to understand the technical content of the present invention, the calculation method of the Vlasov-Poisson equations in high-dimensional space is described in detail below, and the content of the present invention is further explained in conjunction with the accompanying drawings.

[0038] like Figure 1The flowchart of the scheme of the present invention is shown. The technical scheme of the present invention is: a high-precision numerical method for solving the Varasov equation in high-dimensional space using a structured grid, by decomposing the high-dimensional space into two mutually orthogonal subspaces, projecting the Varasov equation into the two subspaces for step-by-step integration and solving. The spectral element method is used to expand the physical quantities in the two subspaces, and the time integration is completed in three substeps in the physical, velocity and physical subspaces. The electrostatic field in the physical subspace is obtained by solving the Poisson equation.

[0039] The specific steps include:

[0040] S1. Analyze the high-dimensional Varasov equation according to the dimension of the target plasma calculation and the shape of the calculation area;

[0041] According to the dimension of the target plasma calculation, the dimensionless Vlasov equation is given as follows:

[0042]

[0043] in, Represents the charged particles in the plasma in phase space The probability distribution function in , d is the dimension of the subspace, d = 1, 2, 3.

[0044] In the calculation, the area is generally selected to be large enough to meet the boundary condition that the probability distribution of charged particles is zero.

[0045] Force Field is the electrostatic field, obtained by solving the following Poisson equation:

[0046] in, is the density distribution function of the charged particle, which is obtained by integrating the probability density distribution function of the charged particle in the velocity space (3).

[0047] is the electrostatic potential distribution function, obtained by solving Poisson's equation (2).

[0048] S2, using hexahedral grids to discretize the physical and velocity subspaces;

[0049] In the six-dimensional phase space calculation, the two subspaces are three-dimensional regions, and the regions are divided using hexahedral units. Consists of physics and velocity subspaces. Divided into M units, each unit is (m=1,2,3,…,M) means, Divided into N units, each unit is (n=1,2,3,…,N) means,

[0050]

[0051] in, and yes and The approximate calculation area is composed of many discrete units. In the embodiment of the present invention, K represents a hexahedral unit, and K+ represents a hexahedral unit adjacent to K.

[0052] Similarly, when calculating in 2P-dimensional phase space, both subspaces are P-dimensional intervals, and the units in P-dimensional space are used. Assume that the calculation area After discreteness, There are a total of M P-dimensional units D m (m=1,2,3,…,M), and There are a total of M units D in P-dimensional space n (n=1,2,3,…,N), then all units can be expressed as

[0053]

[0054] It should be noted that the discretization of the two subspaces is completely independent, and each grid point in the subspace has another complete subspace. Figure 2 As shown, a schematic diagram is given showing that each grid point on a subspace corresponds to a complete other subspace.

[0055] S3, selecting a basis function, and expanding the particle probability density distribution of each unit in the grid of all subspaces and the electrostatic field of each unit in the physical subspace using the basis function;

[0056] The present invention uses a modal expansion basis formed by Jacobi polynomials to expand physical quantities. Jacobi polynomials are characteristic functions of the following Sturm-Liouville equations:

[0057] Where λp=-p(α+β+p+1).

[0058] They have the following orthogonality:

[0059] The C here is related to α, β, and p.

[0060] The modal expansion basis used in the present invention is obtained by vector multiplication of the following one-dimensional modal expansion basis.

[0061] The one-dimensional modal expansion basis is:

[0062] wherein is the Jacobi polynomial when α = β = 1 on the standard interval Ω = {r | -1 < r < 1} The first and last expansion bases are linear to ensure the continuity condition at the cusp.

[0063] The main advantages of the spectral element method (SEM) are its flexibility in dealing with complex geometries and high-order accuracy. Almost all operations and data are local, including interpolation, integration, solving the Poisson equation, and the transformation between physical and spectral spaces, etc. The locality of SEM also makes parallelization easy and the parallel efficiency higher, so the numerical method of this patent adopts SEM.

[0064] Figure 3 Shows a schematic diagram of a hexahedral mesh in a three-dimensional subspace. As can be seen from the figure, there are also subdivided meshes in each hexahedron. They correspond to the order of the modal expansion basis used on the hexahedron. Moreover, these internal subdivided meshes are not uniform. This is due to our adoption of Gauss-Lobatto-Legendre integration points. According to the integral mathematical theory on the interval [-1, 1], using Gauss-Lobatto-Legendre integration points for numerical integration can ensure the exact value for the integration of functions with polynomial order less than P.

[0065] At the same time, the sizes of the eight hexahedrons can be different. The quadrilaterals near the origin are smaller, while those far from the origin are larger. The length and width of the quadrilaterals are also arbitrary.

[0066] Figure 4 Is a schematic diagram of the shapes of 125 = (1 + 4) × (1 + 4) × (1 + 4) modal expansion bases corresponding to the order of 4 of the expansion basis on the hexahedron. They are divided into 8 vertex modes, 36 edge modes, 54 face modes, and 27 internal modes. The vertex mode basis functions are linear functions. They take 1 at a certain vertex and 0 at all other three vertices. The edge mode expansion basis is non-zero only inside one edge and zero at all vertices and the remaining 11 edges. The face mode expansion basis is non-zero only inside one face and zero at all vertices and the remaining 5 faces. The internal mode functions take 0 at all vertices, all edges, and all faces, and are non-zero only at the internal grid points.

[0067] Thus, for the particle probability density distribution function f(x, y, v x , v y , t) in the six-dimensional phase space can be expressed as:

[0068]

[0069] Where M is the number of hexahedral units in the physical subspace, and N is the number of hexahedral units in the velocity subspace. i (x),φ j (y) are the one-dimensional expansion bases in the x and y directions in the physical subspace, and φ k (v x ),φ l (v y ) are respectively v in the velocity subspace x and v y The one-dimensional expansion basis in the direction. The expansion coefficient of the probability density distribution function in the spectral space is shown in Figure 2. It can be seen that if the solution is directly solved in the six-dimensional space, the dimensional challenge is very huge.

[0070] In the present invention, in order to overcome the difficulty of high dimensionality, we only discretize in two three-dimensional subspaces, which greatly reduces the amount of calculation and makes calculation in high-dimensional space possible.

[0071] In actual six-dimensional space calculations, the particle probability density distribution function f(x,y,v x ,v y ,t) is distributed in two subspaces, namely, in the three-dimensional physical subspace (x, y) and the velocity subspace (v x ,v y ) is expanded as follows:

[0072] For each (v x ,v y )∈Ω(v x ,v y ):

[0073]

[0074] For every (x,y)∈Ω(x,y):

[0075]

[0076] From the above expansions (10-11), we can see that they are greatly simplified compared to (9). This is one of the important reasons why it is possible to solve the Vlasov equation in six-dimensional space. Another important reason is parallel computing.

[0077] Even if the high-dimensional space is decomposed into two low-dimensional spaces and multiplied to replace and use high-precision numerical approximation to reduce the degree of freedom of the variables, the computational complexity of the entire algorithm is still very large. In order to further overcome this difficulty, this patent also proposes to use parallel methods on two subspaces at the same time to further accelerate the algorithm. However, since this patent uses a semi-Lagrangian method, it is necessary to track the position of the grid point according to its velocity value at a certain characteristic moment, so parallel communication is required before each time step integration to exchange the situation when the position of the grid point is on a nearby ringing processor. Figure 5 A schematic diagram of the data structure is given in which each processor needs to store the data exchanged with other processors separately when four processors are used in a three-dimensional subspace, and then call the MPI_Alltoall function to exchange data. This is also one of the key technologies that enables the invention of this patent to be realized.

[0077] S4. In space, the conservation conditions are combined to form a semi-discrete equation related to time;

[0078] The first substep is in the physical subspace, and the time integration step is Δt / 2:

[0079] The second substep is in the velocity subspace, and the time integration step is Δt:

[0080] Among them By solving Poisson's equation we obtain:

[0081] The third substep is in the physical subspace, and the time integration step is Δt / 2:

[0082] S5. Solve Poisson's equation in physical space to obtain the electrostatic field at each grid point;

[0083] The Poisson equation in the two-dimensional physical subspace is the above equation (14)

[0084] Multiply both sides of the equation by the three-dimensional expansion basis function and integrate to get:

[0085] in is the three-dimensional expansion basis function, is the electrostatic potential function.

[0086] Formula (17) can be further written as:

[0087] This leads to the following linear system:

[0088] Among them A bb is the coefficient formed by the boundary basis function and the boundary basis function, A ii is the coefficient formed by the internal basis function and the internal basis function, C bi is the coefficient formed by the boundary basis function and the internal basis function, C bi It is the coefficient formed by the internal basis function and the boundary basis function. b is the unknown coefficient on the boundary, φ i is the unknown coefficient of the interior region. b is the right-hand term on the boundary, f b is the right-hand term of the interior region.

[0089] Applying the Shur decomposition method, we can get:

[0090] When the grid points of the three-dimensional physical subspace are very large, the above linear system cannot be solved using a direct solver because the sparse matrix cannot be stored in the memory. The present invention uses an iterative solver of the conjugate gradient method. Since the time step is not large, the change of the electrostatic field is not large, so the present invention uses the result of the previous time step as the initial field and iterates, and generally only a few iterations are required to converge to the required accuracy.

[0091] S6. In terms of time, the implicit SLM method is used to complete the time integration in three steps;

[0092] The semi-Lagrangian method uses the law of conservation of physical quantities along characteristic lines in phase space. At each step, all grid points are first inferred to the positions of the previous half step or one step. They may be on the original processor or on an adjacent processor. If they are not on the same processor, the position information needs to be sent to the processor where it is located through parallel communication. Then each processor calculates the physical quantities of all points on its grid at the previous moment through interpolation operations. Finally, the interpolation results are returned to the processor in need.

[0093] The interpolation operation uses the following expression:

[0094] φ i (x0) is the value of the one-dimensional i-th basis function at x0, φ j (y0) is the value of the j-th one-dimensional basis function at y0, φ k(z0) is the value of the j-th basis function at z0. Since the spectral element method is used, this interpolation operation can be easily completed, and since the accuracy of the interpolation decreases at an exponential rate as the expansion order increases, a very high accuracy can be achieved. This is much more accurate than the general cubic interpolation method. This is also one of the reasons why the present invention adopts the spectral method for calculation.

[0095] In addition, if the interpolation result is negative, it will be forced to 0. This is because the probability density distribution function of the particle cannot be negative.

[0096] Furthermore, during the time integration process, mass conservation needs to be maintained. This is ensured by spatially integrating the entire calculation area after each time integration step. Actual calculations have shown that mass conservation is well maintained due to the high accuracy of the spectral method and the rationality of the algorithm.

[0097] For researchers and technicians in this field, the method of the present invention is also applicable to the calculation of Vlasov-Poisson equations in higher dimensions and more complex regions. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method, characterized in that: By projecting the high-dimensional space onto two mutually orthogonal low-dimensional spaces, and then using the high-order precision spectral element method to discretize the physical variables of the plasma field in the two low-dimensional spaces on a structured grid, the semi-Lagrangian method is used to perform step-by-step time integration in the two low-dimensional spaces respectively, and the time integration of one step in the high-dimensional space is completed through three-step step-by-step integration in the low-dimensional space, and the change process of the plasma physical quantity over time is solved through iteration.

2. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 1 is characterized in that: The specific steps include: S1. Analysis of high-dimensional Vlasov-Poisson equations based on the dimensions and spatial shape of plasma calculations; S2, using a suitable grid to discretize the corresponding subspace; S3, selecting a basis function, and expanding the particle probability density distribution field of each unit and the electrostatic field and current density field on the physical subspace using the basis function; S4. In space, the conservation conditions are combined to form a semi-discrete equation related to time; S5. Solve Poisson's equation in computational physical space to obtain the electrostatic field at each grid point; S6. In time, using the implicit semi-Lagrangian method (SLM, S emi- L agrange M ethod) is completed in three steps for time integration.

3. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 2 is characterized in that: The method for solving the Varasov equation in the high-dimensional space described in step S1 is realized by solving the corresponding diffusion equation in the corresponding physical and velocity low-dimensional spaces and solving the Poisson equation Poisson in the low-dimensional physical space.

4. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 2 is characterized in that: The discrete solution area in step S3 includes the discretization of the physical space velocity space; specifically: the six-dimensional area uses hexahedrons in two three-dimensional spaces as units for discretization.

5. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 2 is characterized in that: Step S4 is specifically as follows: the plasma dynamics equation without particle collision is projected on the physical and velocity subspaces to obtain the corresponding diffusion equation. Each time integration step in the high-dimensional space is completed by three steps in the velocity, physical and velocity low-dimensional spaces respectively.

6. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 2 is characterized in that: Step S5 means that the time integral of the velocity subspace requires the electrostatic field intensity, which is obtained by solving a Poisson equation in the physical space.

7. The numerical method for solving the plasma dynamics equations in high-dimensional space without particle collisions based on the high-order precision spectral element method according to claim 2 is characterized in that: Step S6 means that the time integral in each subspace is implemented by the semi-Lagrangian method. This requires interpolation operations, which can be easily implemented by the spectral element method, and ensures the correctness and accuracy of long-term integration and the conservation of overall mass. In addition, the present invention uses parallel computing technology, which can greatly accelerate the calculation and solve the huge challenges brought by high dimensions.

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