Convergence analysis of coarse mesh finite difference method applied to two-group three-dimensional neutron diffusion problem

Figures & data

Figure 1. Flowchart of CMFD global–local iteration.

Figure 2. Sensitivity of convergence rate to core size (2-N CMFD; 2-G 3D model EVP).

Figure 3. Convergence rate vs. mesh size (2-N CMFD; 2-G 3D model problem).

Figure 4. Sensitivity of 1-N CMFD convergence rate to mesh size and relaxation parameter with 2-G 3D model EVP (Shin's CCF; Jacobi-style Jin update, single sweep). (a) Convergence rate vs. mesh size and (b) convergence rate vs. relaxation parameter.

Figure 5. Sensitivity of 1-N CMFD convergence rate to sweep strategy for the 2-G 3D model EVP (Shin's CCF).

Figure 6. Sensitivity of 1-N CMFD convergence rate to CCF form (Jacobi/G–S-style double sweeps, R–B ordering).

Figure 7. Sensitivity of 1-N CMFD convergence rate to CCF form (Shin's form B: CCF update after all the local 1-N sweeps).

Table 1. Cross section data for 2-G problem.

Group g	Dg	Σrg	νΣfg	Σ12
1	1.500711	0.026968	0.00498277	0.0182498
2	0.415038	0.065255	0.0839026

Table 2. List of test cases (3D cubic core with uniform meshes).

# of meshes	Mesh size (cm)	Core size (cm)
50	2	100
30	3.5	105
20	5	100
14	7.5	105
10	10	100
10	20	200
10	40	400
10	80	800

Table 3. Parameters for 1-N CMFD strategy.

Parameter	Value
CCF form	Shin's form/Joo's form
Jin update	Jacobi/G–S
# of sweeps	1–7
Under-relaxation	0 <  η  < 1
Sweep ordering	Natural/red–black

Figure 8. PARCS quarter-core model for NEACRP A1 case (A: 100 steps, C: 200 steps).

Figure 9. Jacobi vs. G–S Jin update (Joo's CCF, triple sweeps).

Table 4. Summary of NEACRP A1 calculation (Joo's CCF, G–S Jin update, R–B ordering).

(a) Steady state
	2-N	1-N (# of sweeps)
		1	3	5	7
Critical Boron Concentration (CBC: ppm)	561.256	Diverge	561.256
Nodal update	13		35	12	10
CMFD update	36		99	35	30
Nodal time (s)	0.7		5.4	2.9	3.3
CMFD time (s)	1.9		4.3	1.9	1.8
Total time (s)	2.6		9.8	4.9	5.2
Time/nodal sweeps (s)	0.053	0.052	0.049	0.047
(b) Transient
	2-N	1-N (# of sweeps)
		3	5	7
Peak power (%)	125.87	125.86	125.84	125.99
Nodal update	31	49	28	24
CMFD update	274	292	278	270
Nodal time (s)	1.7	7.7	6.7	7.9
CMFD time (s)	19.9	20.5	19.6	19.4
Total time (s)	27.9	34.6	32.9	33.6
Time/nodal sweeps (s)	0.054	0.052	0.048	0.047

Figure 10. Convergence rate of 1-N/2-N CMFD (Joo's CCF, G–S Jin update).

Figure 11. Convergence of single-sweep 1-N CMFD for NEACRP A1 steady state (Joo's CCF, G–S Jin update, R–B ordering). (a) Error reduction vs. relaxation parameter and (b) CCF convergence of node (k = 9, l = 40).

Table 5. Summary of NEACRP A1 steady state with single-sweep 1-N CMFD (Joo's CCF, G–S Jin update, R–B ordering).

	2-N	1-N (relaxation parameter)
		1.0–0.8	0.7	0.6	0.5
CBC (ppm)	561.256	Diverge	561.256
Nodal update	13		89	28	33
CMFD update	36		251	79	91
Nodal time (s)	0.7		6.42	2	2.3
CMFD time (s)	1.9		10.23	3.8	4.1
Total time (s)	2.6		16.7	5.9	6.5
Time/nodal sweeps (s)	0.053		0.072	0.072	0.070

Table 6. Summary of NEACRP A1 steady state with double-sweep 1-N CMFD (Joo's CCF, G–S Jin update, R–B ordering).

	2-N	1-N (relaxation parameter)
		1	0.9	0.8	0.7	0.6
CBC (ppm)	561.256	Diverge	561.256
Nodal update	13		80	23	14	16
CMFD update	36		226	65	40	46
Nodal time (s)	0.7		9.19	2.7	1.6	1.9
CMFD time (s)	1.9		10.58	3.3	2.4	2.5
Total time (s)	2.6		19.83	6.1	4.1	4.5
Time/nodal sweeps (s)	0.053		0.057	0.058	0.056	0.060

Table 7. Summary of NEACRP A1 steady state with triple-sweep 1-N CMFD (Joo's CCF, G–S Jin update, R–B ordering).

	2-N	1-N (relaxation parameter)
		1	0.9	0.8	0.7
CBC (ppm)	561.256	561.256
Nodal update	13	35	17	11	12
CMFD update	36	99	49	33	35
Nodal time (s)	0.7	5.4	2.6	1.7	1.9
CMFD time (s)	1.9	4.3	2.3	1.8	1.9
Total time (s)	2.6	9.8	4.9	3.6	3.8
Time/nodal sweeps (s)	0.053	0.052	0.050	0.052	0.052

Figure 12. Convergence of 1-N CMFD for NEACRP A1 steady state (Joo's CCF, G–S Jin update, R–B ordering).

Table 8. Summary of NEACRP steady state calculations (Joo's CCF; triple sweeps (1-N CMFD), G–S Jin update, R–B ordering, under-relaxation 0.8).

	A1		B1		C1		A2
	HZP, QC		HZP, QC		HZP, HC		HFP, QC
Parameters	2-N	1-N	2-N	1-N	2-N	1-N	2-N	1-N
Nodal update	13	11	11	11	11	19	13	11
CMFD update	36	33	31	32	31	58	37	33
Nodal time (s)	0.7	1.9	0.6	1.7	1.3	5.8	0.8	1.7
Total time (s)	2.8	3.8	2.4	3.8	4.9	12.1	3.5	4.1
Time/nodal sweep (s)	0.055	0.056	0.055	0.052	0.115	0.102	0.058	0.052

Table 9. Summary of NEACRP transient calculations (Joo's CCF; triple sweeps (1-N CMFD), G–S Jin update, R–B ordering, under-relaxation 0.8).

	A1		B1		C1		A2
	HZP, QC		HZP, QC		HZP, HC		HFP, QC
Parameters	2-N	1-N	2-N	1-N	2-N	1-N	2-N	1-N
Peak power (%)	125.9	126.0	125.9	126.0	482.2	482.4	108.6	108.6
Nodal update	31	29	30	28	19	15	10	10
CMFD update	274	277	262	261	327	322	182	208
Nodal time (s)	1.7	4.3	1.8	4.3	2.3	4.5	0.6	1.5
Total time (s)	27.9	29.7	27.3	29.5	81.3	83.3	19.1	20.1
Time/nodal sweep (s)	0.054	0.050	0.060	0.051	0.120	0.101	0.056	0.050