Application of KDF-SPH Method in Numerical Solution of Fractional Convection-diffusion Equation

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

Abstract

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

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.

Figures/Tables 11

References 24

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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

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

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

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

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