KDF-SPH方法在分数阶对流扩散方程数值解中的应用

doi:10.19596/j.cnki.1001-246x.8820

摘要/Abstract

摘要：

将光滑粒子流体动力学(SPH)方法基础上提出的避免核函数导数的SPH (KDF-SPH)方法应用到时间分数阶对流扩散方程的数值求解。在时间分数阶对流扩散方程的模拟计算过程中, 对Caputo时间分数阶导数采用有限差分方法(FDM), 对空间导数分别采用KDF-SPH方法和SPH方法。结果表明: KDF-SPH方法比SPH方法误差小。KDF-SPH保留了SPH (无网格、拉格朗日和粒子性质)的优点, 该方法在减少误差以及保持稳定性方面发挥了较大作用。无论核梯度是否存在, 数值近似都可以进行, 避免了核函数导数的计算, 降低了对核函数可导性的要求, 提高了计算效率, 易于编程实现, 便于扩展高维问题的计算。

关键词: KDF-SPH, 有限差分方法, Caputo分数阶导数, 数值模拟

Abstract:

Based on the smoothed particle hydrodynamics (SPH) method, the SPH method without kernel function derivative (KDF-SPH) is applied to the numerical solution of the time fractional convection-diffusion equation. In the simulation process of the time fractional convection-diffusion equation, the finite difference method (FDM) is used for the Caputo time fractional derivative, and the KDF-SPH method and SPH method are used for the spatial derivative respectively. The results show that the error of KDF-SPH method is much smaller than that of SPH method. Compared with the SPH method, KDF-SPH retains all the advantages of SPH (meshless, Lagrangian and particle properties). This method plays a great role in reducing errors and maintaining stability, and numerical approximation can be carried out regardless of whether the kernel gradient exists or not. It avoids the calculation of the derivative of the kernel function, reduces the requirement for the derivability of the kernel function, improves the calculation efficiency and is easy to be programmed. It is easy to expand the calculation of high-dimensional problems and has good practical application value.

Key words: KDF-SPH, fini, Caputo fractional derivative, numerical simulation

张秀霞, 热合买提江·依明. KDF-SPH方法在分数阶对流扩散方程数值解中的应用[J]. 计算物理, 2025, 42(1): 18-27.

Xiuxia ZHANG, Imin RAHMATJAN. Application of KDF-SPH Method in Numerical Solution of Fractional Convection-diffusion Equation[J]. Chinese Journal of Computational Physics, 2025, 42(1): 18-27.

图/表 11

图1 一维常系数问题在不同时刻的精确解与数值解

Fig.1 Exact solution and numerical solution of one-dimensional constant coefficient problem at different time

表1 一维常系数问题取不同粒子所产生误差范数

Table 1 Error norms generated by different particles in one-dimensional constant coefficient problem

粒子数	SPH方法		KDF-SPH方法
粒子数	L_∞	L₂	L_∞	L₂
21	0.187 824 245	0.029 230 384 9	2.826 246 99×10^-5	4.387 049 88×10^-6
41	0.188 761 281	0.021 279 728 1	5.966 435 60×10^－6	6.713 923 22×10^-7
81	0.189 222 271	0.015 270 329 2	3.915 911 05×10^-7	3.192 811 04×10^-8
161	0.189 461 295	0.010 878 059 2	1.004 535 21×10^-6	5.712 619 91×10^-8

图2 一维变系数问题的在不同时刻的精确解与数值解

Fig.2 Exact solution and numerical solution of one-dimensional variable coefficient problem at different times

表2 一维变系数问题取不同粒子所产生误差范数

Table 2 Error norms generated by different particles in one-dimensional variable coefficient problem

粒子数	SPH方法		KDF-SPH方法
粒子数	L_∞	L₂	L_∞	L₂
21	0.426 023 65	0.065 691 974 5	0.012 750 278 14	0.082 466 900 91
41	0.400 575 14	0.044 381 002 9	0.054 684 664 84	0.006 040 478 08
81	0.386 139 20	0.030 455 404 6	0.040 594 508 69	0.003 174 262 73
161	0.381 061 91	0.024 589 024 6	0.035 873 050 50	0.002 288 587 91

图3 一维Burgers问题在不同时刻的精确解与数值解

Fig.3 Exact solution and numerical solution of one-dimensional Burgers problem at different time

表3 一维Burgers问题取不同粒子所产生误差范数

Table 3 Error norms generated by different particles in one-dimensional Burgers problem

粒子数	SPH方法		KDF-SPH方法
粒子数	L_∞	L₂	L_∞/10^-5	L₂/10^-6
21	0.044 363 838	0.006 787 726 8	5.171 276 14	7.655 307 54
41	0.044 448 379	0.004 916 989 0	4.715 602 54	5.111 371 43
81	0.044 448 970	0.003 519 807 4	4.719 726 09	3.694 452 17
161	0.044 451 803	0.002 504 346 9	4.781 209 45	2.675 855 92

图4 一维对流边界条件问题在不同时刻的精确解与数值解

Fig.4 Exact solution and numerical solution of one-dimensional convection boundary condition problem at different time

表4 一维对流边界问题取不同粒子所产生误差范数

Table 4 Error norms generated by different particles in one-dimensional convection boundary problem

粒子数	SPH方法		KDF-SPH方法
粒子数	L_∞	L₂	L_∞	L₂
21	1.297 222 0	0.202 003 5	0.035 649 2	0.004 942 0
41	1.308 954 7	0.147 337 7	0.035 853 0	0.003 586 2
81	1.313 436 6	0.105 832 2	0.035 969 3	0.002 568 3
161	1.315 674 7	0.075 428 3	0.035 981 6	0.001 827 5

图5 无解析解的问题在不同时刻时浓度扩散图(a) KDF-SPH；(b)Ref.[18]；(c) SPH

Fig.5 Concentration diffusion diagram of problem without analytical solution at different time (a) KDF-SPH; (b) Ref. [18]; (c) SPH

图6 二维问题用不同方法模拟的数值解与精确解(a)y=0.5；(b) x=0.5

Fig.6 Numerical solution and exact solution of two-dimensional problem simulated by different methods (a) y=0.5;(b) x=0.5

表5 二维问题取不同粒子所产生的误差范数

Table 5 Error norms generated by different particles in two-dimensional problem

粒子数	SPH方法		KDF-SPH方法
粒子数	L_∞	L₂	L_∞	L₂
21	0.260 882 9	0.053 923 2	0.020 470 2	0.004 419 4
41	0.484 058 2	0.050 855 9	0.024 548 5	0.002 318 8
81	0.935 894 8	0.049 633 1	0.045 805 4	0.001 991 6
161	1.841 253 9	0.049 071 9	0.091 089 6	0.001 963 8

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