Numerical investigation on a comprehensive high-order finite particle scheme

doi:10.1007/s10483-025-3262-9

Abstract

Abstract:

In the field of discretization-based meshfree/meshless methods, the improvements in the higher-order consistency, stability, and computational efficiency are of great concerns in computational science and numerical solutions to partial differential equations. Various alternative numerical methods of the finite particle method (FPM) frame have been extended from mathematical theories to numerical applications separately. As a comprehensive numerical scheme, this study suggests a unified resolved program for numerically investigating their accuracy, stability, consistency, computational efficiency, and practical applicability in industrial engineering contexts. The high-order finite particle method (HFPM) and corrected methods based on the multivariate Taylor series expansion are constructed and analyzed to investigate the whole applicability in different benchmarks of computational fluid dynamics. Specifically, four benchmarks are designed purposefully from statical exact solutions to multifaceted hydrodynamic tests, which possess different numerical performances on the particle consistency, numerical discretized forms, particle distributions, and transient time evolutional stabilities. This study offers a numerical reference for the current unified resolved program.

Key words: numerical method, high-order finite particle method (HFPM), kernel gradient correction (KGC), decoupled finite particle method (DFPM), weakly compressible smoothed particle hydrodynamics (SPH)

2010 MSC Number:

O175

Yudong LI, Yan LI, Chunfa WANG, P. JOLI, Zhiqiang FENG. Numerical investigation on a comprehensive high-order finite particle scheme. Applied Mathematics and Mechanics (English Edition), 2025, 46(6): 1187-1214.

TrendMD

Figures/Tables 26

Fig. 1

Fig. 2

Table 1

Relative L2-norm errors of different numerical schemes in 2D cases"

Method	First-order	Second-order in form of finite-difference	Second-order in form of indexes
FPM	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$3.459 0 × 10 − 5$ (a)
KGC	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$3.459 0 × 10 − 5$ (a)
DFP	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$3.459 0 × 10 − 5$ (a)
SPH	0.007 8	0.007 8	0.015 5 (a)
HFPM	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$5.417 8 × 10 − 5$ (b)
HKGC	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$1.045 6 × 10 − 4$ (b)
HDFP	$1.027 5 × 10 − 6$	$1.717 5 × 10 − 4$	$0.326 5$ (b)
HSPH	0.007 8	0.007 8	0.565 9 (b)
Note: (a) denotes nested sum, and (b) denotes direct high-order discretization

Table 1

Fig. 3

Table 2

Relative L2-norm errors on different numerical schemes in 3D cases"

Method	First-order	Second-order on form of finite-difference	Second-order on form of indexes
FPM	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	$2.635 7 × 10 − 6$ (a)
KGC	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	$3.425 8 × 10 − 6$ (a)
DFP	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	$3.425 8 × 10 − 6$ (a)
SPH	0.002 5	0.002 5	0.005 1 (a)
HFPM	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	$1.557 7 × 10 − 6$ (b)
HKGC	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	$4.178 0 × 10 − 6$ (b)
HDFP	$1.123 1 × 10 − 7$	$6.808 2 × 10 − 6$	0.681 9 (b)
HSPH	0.002 5	0.002 5	0.994 4 (b)
Note: (a) denotes nested sum, and (b) denotes direct high-order discretization

Table 2

Fig. 4

Table 3

Convergence of relative L2-norm errors on different numerical schemes"

Method	$Δ d$	First-order	Second-order in form of finite-difference	Second-order in form of indexes
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	0.001 5 (a)
FPM	0.01	$3.008 7 × 10 − 5$	0.001 1	$6.195 9 × 10 − 4$ (a)
	0.005	$7.825 2 × 10 − 6$	0.003 8	$1.060 9 × 10 − 4$ (a)
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	0.001 5 (a)
KGC	0.01	$3.008 7 × 10 − 5$	0.001 1	$6.197 0 × 10 − 4$ (a)
	0.005	$7.825 2 × 10 − 6$	0.003 8	$1.063 7 × 10 − 4$ (a)
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	0.001 5 (a)
DFP	0.01	$3.008 7 × 10 − 5$	0.001 1	$6.197 0 × 10 − 4$ (a)
	0.005	$7.825 2 × 10 − 6$	0.003 8	$1.063 7 × 10 − 4$ (a)
	0.05	0.003 3	0.002 8	0.005 4 (a)
SPH	0.01	0.002 6	0.002 9	0.005 0 (a)
	0.005	0.002 5	0.004 6	0.004 8 (a)
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	$8.098 3 × 10 − 4$ (b)
HFPM	0.01	$3.008 7 × 10 − 5$	0.001 1	$2.547 6 × 10 − 4$ (b)
	0.005	$7.825 2 × 10 − 6$	0.003 8	0.001 1 (b)
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	$6.848 5 × 10 − 4$ (b)
HKGC	0.01	$3.008 7 × 10 − 5$	0.001 1	$5.758 7 × 10 − 4$ (b)
	0.005	$7.825 2 × 10 − 6$	0.003 8	0.002 4 (b)
	0.05	$7.509 7 × 10 − 4$	$3.906 1 × 10 − 4$	0.681 8 (b)
HDFP	0.01	$3.008 7 × 10 − 5$	0.001 1	0.681 0 (b)
	0.005	$7.825 2 × 10 − 6$	0.003 8	0.681 9 (b)
	0.05	0.003 3	0.002 8	0.994 3 (b)
HSPH	0.01	0.002 6	0.002 9	0.993 3 (b)
	0.005	0.002 5	0.004 6	0.994 4 (b)
Note: (a) denotes nested sum, and (b) denotes direct high-order discretization

Table 3

Fig. 5

Fig. 6

Table 4

Relative L2-norm errors of different modes of particle distributions"

Method	Mode of particle distribution	First-order	Second-order in form of finite-difference	Second-order in form of indexes
	Uniform	$5.707 6 × 10 − 5$	0.001 6	$2.697 5 × 10 − 4$ (a)
FPM	PST	$1.558 2 × 10 − 4$	0.646 9	0.002 2 (a)
	Arbitrary	0.001 5	12.886 0	0.033 9 (a)
	Uniform	$5.707 6 × 10 − 5$	0.001 6	$2.697 5 × 10 − 4$ (a)
KGC	PST	$1.560 5 × 10 − 4$	1.233 9	0.003 4 (a)
	Arbitrary	0.001 5	23.356 0	0.058 4 (a)
	Uniform	$5.707 6 × 10 − 5$	0.001 6	$2.697 5 × 10 − 4$ (a)
DFP	PST	0.012 6	1.216 3	0.236 7 (a)
	Arbitrary	0.112 4	23.061 0	3.250 0 (a)
	Uniform	0.007 8	0.007 9	0.015 6 (a)
SPH	PST	0.022 7	1.179 0	0.356 7 (a)
	Arbitrary	0.174 3	21.629 0	4.118 4 (a)
	Uniform	$5.707 6 × 10 − 5$	0.001 6	$3.710 6 × 10 − 4$ (b)
HFPM	PST	$5.681 7 × 10 − 5$	0.650 7	$6.712 6 × 10 − 4$ (b)
	Arbitrary	$6.155 5 × 10 − 5$	11.142 0	0.001 8 (b)
	Uniform	$5.707 6 × 10 − 5$	0.001 6	$9.519 7 × 10 − 4$ (b)
HKGC	PST	$5.681 0 × 10 − 5$	1.227 5	0.001 1 (b)
	Arbitrary	$6.008 1 × 10 − 5$	20.086 2	0.001 3 (b)
	Uniform	$5.707 6 × 10 − 5$	0.001 6	0.326 4 (b)
HDFP	PST	0.012 6	1.216 3	1.810 5 (b)
	Arbitrary	0.112 4	23.061 0	15.833 0 (b)
	Uniform	0.007 8	0.007 9	0.565 8 (b)
HSPH	PST	0.022 7	1.179 0	2.090 8 (b)
	Arbitrary	0.174 3	21.629 0	17.180 0 (b)
Note: (a) denotes nested sum, and (b) denotes direct high-order discretization

Table 4

Table 5

Relative L2-norm errors adopting different smoothing kernel functions"

Distribution	Method (kernel)	First-order	Second-order in form of finite-difference	Second-order in form of nested sum
Uniform	FPM (K1)	$1.044 0 × 10 − 7$	$4.172 6 × 10 − 6$	$2.658 2 × 10 − 6$
	FPM (K2)	$1.089 6 × 10 − 7$	$3.824 8 × 10 − 6$	$7.541 8 × 10 − 6$
	KGC (K1)	$1.044 0 × 10 − 7$	$4.172 6 × 10 − 6$	$3.834 1 × 10 − 6$
	KGC (K2)	$1.089 6 × 10 − 7$	$3.824 8 × 10 − 6$	$1.145 2 × 10 − 5$
	DFP (K1)	$1.044 0 × 10 − 7$	$4.172 6 × 10 − 6$	$4.409 5 × 10 − 5$
	DFP (K2)	$1.089 6 × 10 − 7$	$3.824 8 × 10 − 6$	$1.239 5 × 10 − 4$
	SPH (K1)	0.002 5	0.002 5	0.005 1
	SPH (K2)	0.002 3	0.002 3	0.004 6
Arbitrary	FPM (K1)	0.006 2	0.403 7	0.007 1
	FPM (K2)	0.005 4	0.320 7	0.005 8
	KGC (K1)	0.007 8	0.598 9	0.022 6
	KGC (K2)	0.007 3	0.476 0	0.019 5
	DFP (K1)	0.064 6	0.596 7	0.100 7
	DFP (K2)	0.056 6	0.474 5	0.081 9
	SPH (K1)	0.086 5	0.590 1	0.121 4
	SPH (K2)	0.076 0	0.470 9	0.100 5
Note: kernel (K1) denotes quintic kernel, and kernel (K2) denotes cubic spline kernel

Table 5

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 19

Fig. 20

Fig. 21

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