A method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics

Abstract

An embodiment of the present invention provides a method for unconditionally maintaining the strong stability of a hyperbolic conservation law in computational fluid dynamics, comprising: spatially discretizing the hyperbolic conservation law equation in computational fluid dynamics using a preset spatial discretization method to obtain a semi-discrete system of the hyperbolic conservation law; introducing a stabilization term into the semi-discrete system of the hyperbolic conservation law to form a stabilized semi-discrete system of the hyperbolic conservation law; temporally discretizing the stabilized semi-discrete system of the hyperbolic conservation law using an explicit integrating factor Runge-Kutta method to obtain a fully discrete system of the hyperbolic conservation law; and reasonably approximating the exponential function in the explicit integrating factor Runge-Kutta method of the fully discrete system of the hyperbolic conservation law to obtain a fully discrete format of the hyperbolic conservation law that unconditionally maintains strong stability. This method ensures that the hyperbolic conservation law can maintain strong stability at any time step.

Description

Technical Field

[0001] The present invention relates to the field of computational fluid dynamics, and in particular to a method for unconditionally maintaining the strong stability of hyperbolic conservation laws in computational fluid dynamics. Background Art

[0002] Hyperbolic conservation laws in computational fluid dynamics (CFD) are widely used in fields such as fluid mechanics, aerodynamics, and aerospace. Since analytical solutions to these laws often do not exist, they are driving the in-depth research and development of numerical methods. High-order numerical methods are currently a hot topic in computational fluid dynamics and are crucial for efficiently and accurately solving various models in engineering practice.

[0003] As modern industries, particularly aerospace, continue to evolve towards higher, faster, and more precise performance, accurate CFD simulations increasingly rely on high-precision, high-resolution numerical schemes. Traditional methods for maintaining strong stability in hyperbolic conservation law systems in computational fluid dynamics include the Strong-Stability-Preserving (SSP) Runge-Kutta method, the SSP multi-step method, and the SSP implicit and explicit method. If a spatial discretization scheme combined with a first-order explicit Euler method satisfies certain energy stability requirements, the SSP method can ensure that higher-order time discretization also satisfies energy stability, thereby ensuring that the numerical solution is free of numerical oscillations near spatial discontinuities. Explicit time discretization schemes are particularly convenient for practical applications of nonlinear problems. However, stability imposes strict time step restrictions on explicit schemes. Implicit schemes, on the other hand, require solving a system of nonlinear equations at each time step, reducing computational efficiency. Summary of the Invention

[0004] The embodiment of the present invention provides a method for unconditionally maintaining the strong stability of the hyperbolic conservation law in computational fluid dynamics, so that the hyperbolic conservation law can maintain strong stability at any time step.

[0005] To achieve the above objectives, an embodiment of the present invention provides a method for unconditionally maintaining the strong stability of hyperbolic conservation laws in computational fluid dynamics. The method uses computational fluid dynamics (CFD) to simulate the motion of an object in a fluid. When the fluid has strong nonlinear effects such as shock waves and turbulence, the hyperbolic conservation law equations of CFD are processed using the method. The method includes:

[0006] Step 1: spatially discretizing the hyperbolic conservation law equations of computational fluid dynamics using a preset spatial discretization method to obtain a semi-discrete system of the hyperbolic conservation law, wherein the preset spatial discretization method ensures that the semi-discrete system of the hyperbolic conservation law satisfies strong stability when stepping using the forward Euler method;

[0007] Step 2: Introducing a stabilization term into the semi-discrete system of the hyperbolic conservation law to form a stabilized semi-discrete system of the hyperbolic conservation law; wherein the stabilization term includes a stabilization parameter, and the stabilization parameter satisfies a strong stability condition;

[0008] Step 3: Use the explicit integrating factor Runge-Kutta method to time discretize the stabilized semi-discrete system of the hyperbolic conservation law to obtain the fully discrete system of the hyperbolic conservation law;

[0009] Step 4: Make a reasonable approximation to the exponential function in the Runge-Kutta method of the explicit integrating factor of the fully discrete system of the hyperbolic conservation law, and obtain the fully discrete format of the hyperbolic conservation law that unconditionally maintains strong stability.

[0010] The above technical solution has the following beneficial effects: when simulating hyperbolic conservation law problems, strong stability can be maintained at any time step. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0012] Figure 1 This is a flow chart of a method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to an embodiment of the present invention;

[0013] Figure 2 It is the growth function graph of the RK(s,p) method of the present invention. DETAILED DESCRIPTION

[0014] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0015] like Figure 1As shown, in combination with an embodiment of the present invention, a method for unconditionally maintaining the strong stability of the hyperbolic conservation law of computational fluid dynamics is provided. The motion of an object in a fluid is simulated by computational fluid dynamics of fluid mechanics and / or aerodynamics. When the fluid has strong nonlinear effects such as shock waves and turbulence, the hyperbolic conservation law equations of computational fluid dynamics are processed by the method for unconditionally maintaining the strong stability of the hyperbolic conservation law of computational fluid dynamics. The method for unconditionally maintaining the strong stability of the hyperbolic conservation law of computational fluid dynamics includes:

[0016] Step 1: spatially discretizing the hyperbolic conservation law equations of computational fluid dynamics using a preset spatial discretization method to obtain a semi-discrete system of the hyperbolic conservation law, wherein the preset spatial discretization method ensures that the semi-discrete system of the hyperbolic conservation law satisfies strong stability when stepping using the forward Euler method;

[0017] Step 2: Introducing a stabilization term into the semi-discrete system of the hyperbolic conservation law to form a stabilized semi-discrete system of the hyperbolic conservation law; wherein the stabilization term includes a stabilization parameter, and the stabilization parameter satisfies a strong stability condition;

[0018] Step 3: Use the explicit integrating factor Runge-Kutta method to time discretize the stabilized semi-discrete system of the hyperbolic conservation law to obtain the fully discrete system of the hyperbolic conservation law;

[0019] Step 4: Make a reasonable approximation to the exponential function in the Runge-Kutta method of the explicit integrating factor of the fully discrete system of the hyperbolic conservation law, and obtain the fully discrete format of the hyperbolic conservation law that unconditionally maintains strong stability.

[0020] Applying the fully discrete format of hyperbolic conservation laws that unconditionally maintain strong stability to hyperbolic conservation law models in fluid mechanics, aerodynamics, and aerospace, such as the solution and calculation of Euler equations, can greatly improve computational efficiency.

[0021] Preferably, in step 1, the hyperbolic conservation law equation is in the form of:

[0022]

[0023] Among them, t represents the time coordinate, T represents the end time, u represents the conserved quantity, and u t represents the time derivative of the conserved quantity, represents the space vector coordinate, express Regarding the divergence of spatial coordinates, Ω represents a one-dimensional, two-dimensional, or three-dimensional region, represents the convective vector flux;

[0024] When computational fluid dynamics evolves in a one-dimensional space, take a one-dimensional interval Ω=[xl ,x r ], where x l is the left endpoint of the interval, x r As the right endpoint of the interval, the interval is divided into equal intervals using the spatial step size h. After the division, the number of spatial grid nodes is

[0025] Let u=[u0,u1,…,u N-1 ] T is the semi-discrete solution, and the convection term in Eq. (1) In one-dimensional case, we take f(u) x , the set of ordinary differential equations of the semi-discrete system obtained by spatially discretizing Equation (1) using the preset spatial discretization method is:

[0026] u t =F(u) (2)

[0027] The preset spatial discretization methods include: upwind format or MUSCL format;

[0028] When using the preset spatial discretization method to spatially discretize Equation (1), the resulting semi-discrete system of hyperbolic conservation laws (2) is required to satisfy the strong stability inequality when using the forward Euler method step:

[0029] ‖u n +τF(u n )‖≤‖u n ‖, where 0<τ≤τ FE (3)

[0030] Among them, ‖·‖ can be any norm, semi-norm or convex functional, u n Indicates t n The numerical solution at time τ represents the time step, τ FE is the maximum time step that makes equation (3) valid;

[0031] When the semi-discrete system (2) of hyperbolic conservation laws satisfies the strong stability inequality (3) when stepping using the forward Euler method, the numerical solution of the semi-discrete system (2) is guaranteed to be strongly stable and non-divergent.

[0032] Preferably, in step 2, the parameter and The following conditions are met:

[0033]

[0034] Multiply both sides of equation (3) by get:

[0035]

[0036] It can be seen that it contains parameters Formula (5) satisfies strong stability;

[0037] Will As a stabilizing term, the stabilizing term is introduced into the semi-discrete system (2) The equivalent stabilized form of the ordinary differential equations (2) of the semi-discrete system is obtained:

[0038]

[0039] Formula (6) is regarded as a stabilized semi-discrete system of hyperbolic conservation laws.

[0040] Preferably, in step 3, the explicit integrating factor Runge-Kutta method is used to time discretize the stabilized semi-discrete system (6) of the hyperbolic conservation law to obtain the fully discrete system sIFRK of the hyperbolic conservation law:

[0041]

[0042] Among them, n represents the n-th layer time step, c i represents the node of the Runge-Kutta method, a i,j represents the coefficient of the Runge-Kutta method, τ represents the time step, and s represents the series of the Runge-Kutta method;

[0043] Assumption c i ≥c j , a i,j ≥0, i>j, take the norm ‖·‖ on both sides of formula (7), and then substitute formula (5) into it to obtain:

[0044]

[0045] Where i = 1,…,s;

[0046] Will Defined as a growth function, when the growth function satisfies the following conditions:

[0047]

[0048] We get: ‖u n,i ‖≤‖u n ‖, i = 1,…, s;

[0049] Using mathematical induction, we can deduce that Equation (7) satisfies strong stability: ‖u n+1 ‖≤‖u n ‖≤…≤‖u 0 ‖.

[0050] Preferably, the Butcher table of the Runge-Kutta method with explicit integrating factors satisfying equation (8) is as follows:

[0051]

[0052] In RK(s,p), s represents the series of the Runge-Kutta method, and p represents the order.

[0053] Preferably, in step 4, the exponential function in (7) is recursively calculated. Approximation is performed to eliminate the exponential function in step 3 The exponential decay effect of The approximation is performed as follows:

[0054]

[0055] Exponential function approximated recursively The fully discrete system isIFRK of the hyperbolic conservation law expressed by the improved integrating factor Runge-Kutta method is obtained:

[0056]

[0057] When the coefficients of the Runge-Kutta method in equation (9) satisfy And when the parameter When condition (4) in step 2 is met, formula (8) is expressed as When τ > 0, Equation (9) maintains strong stability for any time step τ > 0, and Equation (9) eliminates the exponential decay effect.

[0058] Preferably, the parameter Satisfying condition (4) in step 2, formula (8) is expressed as When , Equation (9) maintains strong stability for any time step τ>0. The proof of the above conclusion is as follows:

[0059] Taking the norm ‖·‖ on both sides of equation (9), we can obtain the following by mathematical induction:

[0060]

[0061] Where i = 1,…,s

[0062] We get: ‖u n,i ‖ is never greater than ‖u n ‖, so u n+1 Maintain strong stability.

[0063] Preferably, the hyperbolic conservation law equation is formulated as a constant coefficient convection equation u with periodic boundary conditions. t +Cu x = 0, the first-order upwind difference scheme is used to discretize the constant coefficient convection equation with periodic boundary conditions, and the semi-discrete system of hyperbolic conservation law equations is obtained:

[0064] u t =F(u) (2)

[0065] Where C represents a constant coefficient, C>0, F(u)=-CDu, and D is the first-order differential operator. The differential matrix is ​​obtained by discretizing with h as the spatial step size:

[0066]

[0067] Considering the total variation of u According to the total variation of u: When, u satisfies ‖u+τF(u)‖≤‖u‖.

[0068] The above technical solutions of the embodiments of the present invention are described in detail below with reference to specific application examples. For technical details not introduced during the implementation process, please refer to the relevant description above.

[0069] The present invention is a time-discrete format that unconditionally maintains strong stability based on the improved integrating factor Runge-Kutta method. When applied to hyperbolic conservation law models in fluid mechanics, aerodynamics, and aerospace, such as the solution and calculation of Euler equations, the present invention addresses the deficiency that existing high-order implicit formats and explicit formats cannot unconditionally maintain strong stability. By introducing a stabilization term in the semi-discrete equation and using recursive technology to approximate the exponential function, the numerical format can maintain strong stability at any time step, thereby improving the computational efficiency of hyperbolic conservation law problems and being able to be used to simulate complex computational fluid dynamics models.

[0070] 1. A method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics, comprising the following steps:

[0071] Step 1: spatially discretize the hyperbolic conservation law equations of computational fluid dynamics using a preset spatial discretization method to obtain a semi-discrete system of the hyperbolic conservation law, wherein the preset spatial discretization method enables the semi-discrete system of the hyperbolic conservation law to satisfy strong stability when stepping using the forward Euler method;

[0072] Step 2: Introducing a stabilization term into the semi-discrete system of the hyperbolic conservation law to form a stabilized semi-discrete system of the hyperbolic conservation law; wherein the stabilization term includes a stabilization parameter, and the stabilization parameter satisfies a strong stability condition;

[0073] Step 3: Use the explicit integrating factor Runge-Kutta method to time discretize the stabilized semi-discrete system of the hyperbolic conservation law to obtain the fully discrete system of the hyperbolic conservation law;

[0074] Step 4: A reasonable approximation is made to the exponential function of the explicit integrating factor Runge-Kutta method for the fully discrete system of the hyperbolic conservation law to obtain a fully discrete format of the hyperbolic conservation law that unconditionally maintains strong stability.

[0075] The fully discrete format that unconditionally maintains strong stability is applied to hyperbolic conservation law models in fluid mechanics, aerodynamics, and aerospace, such as the solution and calculation of Euler equations.

[0076] In fluid mechanics and aerodynamics simulations, strong nonlinear effects such as shock waves and turbulence are often encountered. When numerically solving the Euler equations, traditional time-stepping schemes impose strict restrictions on the time step size to maintain strong stability, making this a bottleneck restricting the development and application of numerical simulations. Applying a fully discrete scheme that unconditionally maintains strong stability to solve the Euler equations allows for increasing the time step size while maintaining stability, reducing computational time and significantly improving computational efficiency.

[0077] 2. In step 1, the general form of the hyperbolic conservation law equation is:

[0078]

[0079] Among them, t represents the time coordinate, T represents the end time, u represents the conserved quantity, and u t represents the time derivative of the conserved quantity, represents the space vector coordinate, express Regarding the divergence of spatial coordinates, Ω represents a one-dimensional, two-dimensional, or three-dimensional region, represents the convection vector flux; the general form refers to the generality of f(u), and an appropriate f(u) can be chosen arbitrarily.

[0080] When computational fluid dynamics evolves in a one-dimensional space, take a one-dimensional interval Ω=[x l ,x r ], where x l is the left endpoint of the interval, x r As the right endpoint of the interval, the interval is divided into equal intervals using the spatial step size h. After the division, the number of spatial grid nodes is

[0081] Let u=[u0,u1,…,u N-1 ] Tis the semi-discrete solution (the solution of the semi-discrete system), the convection term in Equation (1) can be taken as f(u) x , apply the preset spatial discretization method (1) to perform spatial discretization, and obtain the ordinary differential equations of the semi-discrete system:

[0082] u t =F(u) (2)

[0083] The preset spatial discretization methods include but are not limited to: upwind format or MUSCL format;

[0084] When the preset spatial discretization method is used to spatially discretize Equation (1), the resulting semi-discrete system (2) is required to satisfy the following strong stability inequality when using the forward Euler method step:

[0085] ‖u n +τF(u n )‖≤‖u n ‖, where 0<τ≤τ FE (3)

[0086] Among them, ‖·‖ can be any norm, semi-norm or convex functional, u n Indicates t n The numerical solution at time τ represents the time step, τ FE is the maximum time step that makes equation (3) valid;

[0087] When the semi-discrete system (2) satisfies the strong stability inequality (3) when stepping using the forward Euler method, it can be guaranteed that the numerical solution of the semi-discrete system (2) is strongly stable and does not diverge.

[0088] 3. In step 2, introduce the parameter and The following conditions are met:

[0089]

[0090] Multiply both sides of equation (3) by get:

[0091]

[0092] It is explained by formula (5) The properties of can satisfy strong stability;

[0093] Formulas (4) and (5) are used to give The restriction of provides conditions for the subsequent proof of strong stability;

[0094] Will As a stabilizing term, by introducing the stabilizing term into the semi-discrete system (2) The stabilized form equivalent to the semi-discrete ordinary differential equation system (2) is obtained:

[0095]

[0096] Formula (6) is regarded as a stabilized semi-discrete system of hyperbolic conservation laws.

[0097] 4. In step 3, the explicit integrating factor Runge-Kutta (IFRK) method is used to time discretize the stabilized semi-discrete system (6) of the hyperbolic conservation law to obtain the fully discrete system sIFRK of the hyperbolic conservation law:

[0098]

[0099] Note format (7) as stabilized IFRK (sIFRK);

[0100] Among them, u n,0 Indicates u n The notation (i.e. the first formula of (7), which does not need to be explained), n represents the nth layer time step, c i represents the node of the Runge-Kutta method, a i,j represents the coefficient of the Runge-Kutta method, τ represents the time step, and s represents the number of stages of the Runge-Kutta method;

[0101] Assumption c i ≥c j , a i,j ≥0, i>j, take the norm ‖·‖ on both sides of formula (7) and substitute formula (5) to obtain:

[0102]

[0103] Where i = 1,…,s;

[0104] Will It is defined as a growth function, so it only needs to satisfy the following conditions:

[0105]

[0106] We get: ‖u n,i ‖≤‖u n ‖, i = 1,…, s;

[0107] Using mathematical induction, we can deduce that Equation (7) satisfies strong stability: ‖u n+1 ‖≤‖u n‖≤…≤‖u 0 ‖.

[0108] However, there are few Runge-Kutta methods that satisfy formula (8). The Butcher table of some Runge-Kutta methods that satisfy formula (8) is as follows:

[0109]

[0110] In RK(s,p), s represents the series of the Runge-Kutta method, and p represents the order.

[0111] Due to the attenuation effect of the exponential function, the growth functions of the four Runge-Kutta methods will quickly decay to 0, such as Figure 2 As shown, the numerical solution degenerates to zero solution when the time step is large.

[0112] 5. In step 4, the exponential function in (7) is recursively calculated. Approximation is performed to eliminate the exponential function in step 3 The exponential decay effect of , let:

[0113]

[0114] Exponential function approximated recursively The fully discrete system of hyperbolic conservation laws expressed by the improved integrating factor Runge-Kutta method (improved sIFRK, isIFRK) is obtained:

[0115]

[0116] When the coefficients of the Runge-Kutta method satisfy It can be proved that when the parameter When the selection satisfies the condition (4) in step 2, the format (8) can maintain strong stability for any time step τ>0, because the growth function satisfies

[0117] Therefore, the exponential decay effect is eliminated at the same time.

[0118] When the parameter When the selection of satisfies condition (4) in step 2, format (9) maintains strong stability for any time step τ>0. The proof is as follows: Taking the norm ‖·‖ on both sides of formula (9), we can obtain by mathematical induction:

[0119]

[0120] Where i = 1,…,s

[0121] We get: ‖u n,i ‖ is never greater than ‖u n ‖, so u n+1 Maintain strong stability;

[0122] Improved Runge-Kutta methods of order 1 to 4 include:

[0123]

[0124] In RK(s,p), s represents the series of the Runge-Kutta method and p represents the order. The coefficients of the improved Runge-Kutta method from the first to the fourth order satisfy

[0125] The time discrete format provided by the present invention, which unconditionally maintains strong stability based on the improved integrating factor method, can perform numerical simulations on one-dimensional, two-dimensional, and three-dimensional conservation-type hyperbolic conservation law equations. The simulation method is explicit and high-order, which can greatly improve the computational efficiency and accuracy, and can maintain strong stability for any time step.

[0126] Also, first, in step 1, when computational fluid dynamics evolves in a one-dimensional space, take a one-dimensional interval Ω = [x l ,x r ], consider the constant coefficient convection equation u with periodic boundary conditions on a one-dimensional interval t +Cu x =0, constant coefficient C>0, the first-order upwind difference scheme is used to discretize the constant coefficient convection equation with periodic boundary conditions, and the semi-discrete system of hyperbolic conservation law equations is obtained:

[0127] u t =F(u) (2)

[0128] Where F(u) = -CDu, C represents a constant coefficient, and D is the first-order differential operator with h as the spatial step size. Discretize to get the differential matrix:

[0129]

[0130] Considering the total variation (TV) of u According to the total variation of u, it can be proved that when When , ‖u+τF(u)‖≤‖u‖ is satisfied; total variation is one of the strong stability.

[0131] Second, add a stabilization term for the semi-discrete system as follows:

[0132] Select Parameters The stability term is introduced in formula (2) The stabilized semi-discrete system is obtained:

[0133]

[0134] Third, establish the Runge-Kutta method for stabilizing the integrating factor of the semi-discrete system

[0135] For example, based on RK(4,4), s = 4, and the integral factor Runge-Kutta method is applied to (6) to obtain the fully discrete format

[0136]

[0137] 4. Make a reasonable approximation to the exponential function and establish the improved integral factor Runge-Kutta method, specifically:

[0138] For the exponential function in the fully discrete format (7) Perform recursive approximation and let:

[0139]

[0140] The improved integrating factor Runge-Kutta method with unconditional strong stability is obtained:

[0141]

[0142] The present invention proposes a construction of an explicit time discretization format that unconditionally maintains strong stability for the hyperbolic conservation law system in computational fluid dynamics, by introducing a stabilization term into the hyperbolic conservation law equation after spatial semi-discretization, a type of time discretization improvement processing, especially the selection of stabilization parameters and the approximation of the exponential function. The advantage of the present invention is that the time step can be arbitrarily selected and strong stability can be maintained. It can also be combined with a variety of spatial discretization formats and is suitable for efficient and high-precision numerical simulation of complex flows based on hyperbolic conservation laws. The scheme of the numerical method for unconditionally maintaining strong stability for hyperbolic conservation law problems of the present invention is suitable for increasingly realistic engineering problems, increasingly complex calculation processes, and increasingly high requirements for the calculation effects of models and algorithms (high calculation efficiency and strong stability terms) in the fields of fluid mechanics, aerodynamics, aerospace, etc., and has important practical significance for improving the theoretical basis of the algorithm and broadening its application field.

[0143] It should be understood that the specific order or hierarchy of steps in the disclosed processes is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The accompanying method claims present elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy described.

[0144] In the foregoing detailed description, various features are grouped together in a single embodiment to simplify the disclosure. This method of disclosure should not be interpreted as reflecting an intention that embodiments of the claimed subject matter require more features than are expressly recited in each claim. On the contrary, as reflected in the appended claims, the invention comprises less than all the features of any individual disclosed embodiment. The appended claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.

[0145] The above description of the disclosed embodiments is intended to enable any person skilled in the art to implement or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be applied to other embodiments without departing from the spirit and scope of the present disclosure. Therefore, the present disclosure is not limited to the embodiments presented herein but is intended to be accorded the widest scope consistent with the principles and novel features disclosed herein.

[0146] The foregoing description includes examples of one or more embodiments. Of course, it is not possible to describe all possible combinations of components or methods for the purposes of describing the above embodiments, but one of ordinary skill in the art will recognize that the various embodiments may be further combined and arranged. Therefore, the embodiments described herein are intended to encompass all such changes, modifications and variations that fall within the scope of the appended claims. Furthermore, to the extent the term "comprising" is used in the specification or claims, the term is intended to be encompassed in a manner similar to the term "including," as explained in terms of "including," used as a transitional word in the claims. Furthermore, any use of the term "or" in the specification of the claims is intended to mean a "non-exclusive or."

[0147] Those skilled in the art will also appreciate that the various illustrative logical blocks, units, and steps listed in the embodiments of the present invention can be implemented by electronic hardware, computer software, or a combination of the two. To clearly demonstrate the interchangeability of hardware and software, the various illustrative components, units, and steps described above have generally described their functions. Whether such functions are implemented by hardware or software depends on the specific application and the design requirements of the entire system. Those skilled in the art may use various methods to implement the described functions for each specific application, but such implementation should not be understood as exceeding the scope of protection of the embodiments of the present invention.

[0148] The various illustrative logic blocks or units described in the embodiments of the present invention can be implemented or operated by a general-purpose processor, a digital signal processor, an application-specific integrated circuit (ASIC), a field programmable gate array or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof. The general-purpose processor can be a microprocessor, and optionally, the general-purpose processor can also be any conventional processor, controller, microcontroller or state machine. The processor can also be implemented by a combination of computing devices, such as a digital signal processor and a microprocessor, a plurality of microprocessors, one or more microprocessors combined with a digital signal processor core, or any other similar configuration.

[0149] The steps of the methods or algorithms described in the embodiments of the present invention may be directly embedded in hardware, a software module executed by a processor, or a combination of the two. The software module may be stored in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, a hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art. For example, the storage medium may be connected to the processor so that the processor can read information from the storage medium and write information to the storage medium. Alternatively, the storage medium may also be integrated into the processor. The processor and storage medium may be provided in an ASIC, which may be provided in a user terminal. Alternatively, the processor and storage medium may also be provided in different components in the user terminal.

[0150] In one or more exemplary designs, the above-mentioned functions described in the embodiments of the present invention can be implemented in hardware, software, firmware, or any combination of the three. If implemented in software, these functions can be stored on a computer-readable medium or transmitted in the form of one or more instructions or codes on a computer-readable medium. Computer-readable media include computer storage media and communication media that facilitate the transfer of computer programs from one location to another. Storage media can be any available medium that can be accessed by a general or special computer. For example, such computer-readable media can include but are not limited to RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store program code in the form of instructions or data structures and other forms that can be read by a general or special computer, or a general or special processor. In addition, any connection can be appropriately defined as a computer-readable medium. For example, if the software is transmitted from a website, server or other remote resource via a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless methods such as infrared, wireless, and microwave, it is also included in the definition of computer-readable media. The disks and discs mentioned above include compact disks, laser disks, optical disks, DVDs, floppy disks, and Blu-ray discs. Disks typically reproduce data magnetically, while discs typically reproduce data optically with lasers. Combinations of the above may also be included in computer-readable media.

[0151] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics, characterized in that: In aerospace, computational fluid dynamics (CFD) of fluid mechanics and / or aerodynamics is used to simulate the motion of a space object in a fluid, constructing a hyperbolic conservation law equation for the computational fluid dynamics corresponding to the space object. The analytic solution of the motion of the space object in the fluid is obtained by processing the hyperbolic conservation law equation for the computational fluid dynamics. When the fluid has strong nonlinear effects such as shock waves and turbulence, the hyperbolic conservation law equation for the computational fluid dynamics is processed using the method for unconditionally maintaining the strong stability of the hyperbolic conservation law for the computational fluid dynamics. The method for unconditionally maintaining the strong stability of the hyperbolic conservation law for the computational fluid dynamics includes: Step 1: spatially discretizing the hyperbolic conservation law equations of computational fluid dynamics using a preset spatial discretization method to obtain a semi-discrete system of the hyperbolic conservation law, wherein the preset spatial discretization method ensures that the semi-discrete system of the hyperbolic conservation law satisfies strong stability when stepping using the forward Euler method; Step 2: Introducing a stabilization term into the semi-discrete system of the hyperbolic conservation law to form a stabilized semi-discrete system of the hyperbolic conservation law; wherein the stabilization term includes a stabilization parameter, and the stabilization parameter satisfies a strong stability condition; Step 3: Use the explicit integrating factor Runge-Kutta method to time discretize the stabilized semi-discrete system of the hyperbolic conservation law to obtain the fully discrete system of the hyperbolic conservation law; Step 4: Approximate the exponential function of the explicit integral factor Runge-Kutta method of the fully discrete system of the hyperbolic conservation law reasonably to obtain the fully discrete format of the hyperbolic conservation law that unconditionally maintains strong stability; In step 1, the hyperbolic conservation law equation is in the form of: Among them, t represents the time coordinate, T represents the end time, u represents the conserved quantity, and u t represents the time derivative of the conserved quantity, represents the space vector coordinate, express Regarding the divergence of spatial coordinates, Ω represents a one-dimensional, two-dimensional, or three-dimensional interval, represents the convective vector flux; When computational fluid dynamics evolves in a one-dimensional space, take a one-dimensional interval Ω=[x l ,x r ], where x l is the left endpoint of the interval, x r As the right endpoint of the interval, the interval is divided into equal intervals using the spatial step size h. After the division, the number of spatial grid nodes is Let u=[u0,u1,…,u N-1 ] T is the semi-discrete solution, and in formula (1) In one-dimensional case, we take f(u) x , the set of ordinary differential equations of the semi-discrete system obtained by spatially discretizing Equation (1) using the preset spatial discretization method is: in t =F(u) (2) Among them, the preset spatial discretization methods include: upwind format or MUSCL format; F(u) = -CDu, C represents a constant coefficient, C>0, D is a first-order differential operator The differential matrix is ​​obtained by discretizing with h as the spatial step size: Considering the total variation of u According to the total variation definition of u, it can be proved that: When, u satisfies ‖u+τF(u)‖≤‖u‖; When using the preset spatial discretization method to spatially discretize Equation (1), the resulting semi-discrete system of hyperbolic conservation laws (2) is required to satisfy the strong stability inequality when using the forward Euler method step: ‖u n +τF(u n )‖≤‖u n ‖, in which, 0<τ≤τ FE (3) Among them, ‖·‖ can be any norm, semi-norm or convex functional, u n Indicates t n The numerical solution at time τ represents the time step, τ FE is the maximum time step that makes equation (3) valid; When the semi-discrete system (2) of hyperbolic conservation laws satisfies the strong stability inequality (3) when stepping using the forward Euler method, the numerical solution of the semi-discrete system (2) is guaranteed to be strongly stable and non-divergent.

2. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 1, characterized in that: In step 2, the parameter and The following conditions are met: Multiply both sides of equation (3) by get: It can be seen that it contains parameters Formula (5) satisfies strong stability; Will As a stabilizing term, the stabilizing term is introduced into the semi-discrete system (2) The equivalent stabilized form of the ordinary differential equations (2) of the semi-discrete system is obtained: Formula (6) is regarded as a stabilized semi-discrete system of hyperbolic conservation laws.

3. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 2, characterized in that: In the step 3, the explicit integrating factor Runge-Kutta method is used to time discretize the stabilized semi-discrete system (6) of the hyperbolic conservation law to obtain the fully discrete system sIFRK of the hyperbolic conservation law: Among them, n represents the n-th layer time step, c i represents the node of the Runge-Kutta method, a i,j represents the coefficient of the Runge-Kutta method, τ represents the time step, and s represents the series of the Runge-Kutta method; Assumption c i ≥c j , a i,j ≥0, i>j, take the norm ‖·‖ on both sides of formula (7), and then substitute formula (5) into it to obtain: Where i = 1,…,s; Will Defined as a growth function, when the growth function satisfies the following conditions: Reached: ‖u n,i ‖≤‖u n ‖, i = 1,...,s; Using mathematical induction, we can deduce that Equation (7) satisfies strong stability: ‖u n+1 ‖≤‖u n ‖≤…≤‖u 0 ‖.

4. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 3, characterized in that: The Butcher table of the Runge-Kutta method with explicit integration factors that satisfies equation (8) is as follows: In RK(s,p), s represents the series of the Runge-Kutta method, and p represents the order.

5. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 3, characterized in that: In step 4, the exponential function in (7) is recursively calculated. Approximation is performed to eliminate the exponential function in step 3 The exponential decay effect of The approximation is performed as follows: Exponential function approximated recursively The fully discrete system isIFRK of the hyperbolic conservation law expressed by the improved integrating factor Runge-Kutta method is obtained: When the coefficients of the Runge-Kutta method in equation (9) satisfy And when the parameter When condition (4) in step 2 is met, formula (8) is expressed as When τ > 0, Equation (9) maintains strong stability for any time step τ > 0, and Equation (9) eliminates the exponential decay effect.

6. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 5, characterized in that: The parameters Satisfying condition (4) in step 2, formula (8) is expressed as When , Equation (9) maintains strong stability for any time step τ>

0. The proof of Equation (9) is as follows: Taking the norm ‖·‖ on both sides of equation (9), we can obtain the following by mathematical induction: Where i = 1,…,s We get: ‖u n,i ‖ is never greater than ‖u n ‖, so u n+1 Maintain strong stability.

7. The method for unconditionally maintaining strong stability of hyperbolic conservation laws in computational fluid dynamics according to claim 1, characterized in that: The hyperbolic conservation law equation is transformed into a constant coefficient convection equation with periodic boundary conditions u t +Cu x = 0, the first-order upwind difference scheme is used to discretize the constant coefficient convection equation with periodic boundary conditions, and the semi-discrete system of hyperbolic conservation law equations is obtained: in t =F(u) (2).

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