Advances in Numerical Analysis

Research Article

A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation

Table 1

Experimental error results for Experiment 1.
ℎ ‖ 𝑢 − 𝑢 ℎ ‖ 𝐿 ∞ HOC4cvge. rate ‖ 𝑢 − 𝑢 ℎ ‖ 𝐿 ∞ HOC 6 1 cvge. rate
1/4 1 . 7 8 5 5 6 2 ⋅ 1 0 − 4 3 . 4 9 2 4 1 2 ⋅ 1 0 − 6
1/8 1 . 1 0 4 2 0 8 ⋅ 1 0 − 5 4.02 5 . 7 0 3 7 7 9 ⋅ 1 0 − 8 5.94
1/16 7 . 0 1 6 0 5 4 ⋅ 1 0 − 7 3.98 9 . 0 9 5 8 8 5 ⋅ 1 0 − 1 0 5.97
1/32 4 . 4 3 9 1 3 5 ⋅ 1 0 − 8 3.98 1 . 4 2 1 6 4 3 ⋅ 1 0 − 1 1 6.00
1/64 2 . 7 7 5 5 2 7 ⋅ 1 0 − 9 4.00 2 . 2 2 8 4 7 8 ⋅ 1 0 − 1 3 6.00
HOC 6 2 ( 𝛽 1 9 = 0 . 0 ) rate HOC6 ( 𝛽 7 = 0 . 0 ) rate
1/4 7 . 0 4 8 3 7 2 ⋅ 1 0 − 6 1 . 8 8 4 8 4 9 ⋅ 1 0 − 5
1/8 1 . 1 9 4 9 5 1 ⋅ 1 0 − 7 5.88 3 . 0 4 3 2 2 8 ⋅ 1 0 − 7 5.95
1/16 1 . 8 8 1 6 1 1 ⋅ 1 0 − 9 5.99 4 . 9 1 4 8 5 4 ⋅ 1 0 − 9 5.95
1/32 2 . 9 4 9 3 2 7 ⋅ 1 0 − 1 1 6.00 7 . 7 1 1 2 8 5 ⋅ 1 0 − 1 1 5.99
1/64 4 . 6 2 2 1 8 8 ⋅ 1 0 − 1 3 6.00 1 . 2 0 3 9 3 3 ⋅ 1 0 − 1 2 6.00
HOC6 ( 𝛽 7 = 𝛽 1 9 ) rate HOC6 ( 𝛽 7 = 2 𝛽 1 9 ) rate
1/4 1 . 0 2 1 6 2 0 ⋅ 1 0 − 5 5 . 9 0 0 0 5 9 ⋅ 1 0 − 6
1/8 1 . 6 8 2 9 1 6 ⋅ 1 0 − 7 5.92 1 . 0 0 2 7 6 0 ⋅ 1 0 − 7 5.88
1/16 2 . 6 7 7 3 6 6 ⋅ 1 0 − 9 5.97 1 . 5 7 2 3 4 1 ⋅ 1 0 − 9 6.00
1/32 4 . 2 0 2 4 9 8 ⋅ 1 0 − 1 1 5.99 2 . 4 7 5 5 3 7 ⋅ 1 0 − 1 1 5.99
1/64 6 . 5 6 7 0 3 9 ⋅ 1 0 − 1 3 6.00 3 . 8 5 8 1 6 4 ⋅ 1 0 − 1 3 6.00