Table 1.
Numerical verification of the partition equation.
| Test | Σ | H(B, 𝒱a) | H(B, 𝒱b) | δH | Hsum | H(B, 𝒱) | ϵ | bcAp | bcA | dVCAp | dVCA |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | z = 1 | 6.4022 | 0.3675 | 0.3887 | 7.1584 | 7.2069 | 0.67 | t t b | t t b | y n y | n y n |
| 2 | z = 1 | 6.4101 | 0.3767 | 0.3756 | 7.1625 | 7.2052 | 0.59 | t t b | t t b | y y y | y y y |
| 3 | z = 1 | 6.3943 | 0.3582 | 0.3988 | 7.1514 | 7.2087 | 0.79 | t t b | t t b | n n n | n n n |
| 4 | z = 1 | 6.4101 | 0.3771 | 0.3764 | 7.1637 | 7.2052 | 0.58 | t t t | t t b | y y y | y y y |
| 5 | z = 1 | 6.4045 | 0.3672 | 0.3476 | 7.1193 | 7.1730 | 0.75 | b b t | t t b | n n n | n n n |
| 6 | z = 1 | 6.4218 | 0.3771 | 0.3443 | 7.1433 | 7.1840 | 0.57 | b b t | t t b | y y y | y y y |
| 7 | z = 1 | 6.3943 | 0.3582 | 0.3988 | 7.1514 | 7.2087 | 0.79 | t t b | t t b | n n n | n n n |
| 8 | z = 1 | 6.4220 | 0.3767 | 0.2638 | 7.0625 | 7.2052 | 1.98 | t t b | t b b | y y y | y y y |
| 9 | z = 1 | 6.4047 | 0.3582 | 0.2852 | 7.0481 | 7.2087 | 2.23 | t t b | t b b | n n n | n n n |
| 10 | x = 0 | 0.9215 | 1.0414 | 5.2347 | 7.1976 | 7.2052 | 0.11 | t t b | t t b | y y y | y y y |
| 11 | x = 0 | 0.9215 | 0.9638 | 5.3095 | 7.1948 | 7.2052 | 0.14 | t t t | t t b | y y y | y y y |
| 12 | x = 0 | 0.9215 | 1.0414 | 5.2347 | 7.1976 | 7.2052 | 0.11 | t t b | t t b | y y y | y y y |
| 13 | x = 0 | 0.9662 | 1.0414 | 5.1834 | 7.1910 | 7.2052 | 0.20 | t t b | t b b | y y y | y y y |
| 14 | x = 0 | 0.9720 | 0.7803 | 5.3074 | 7.0596 | 7.2087 | 2.07 | t t b | t b b | n n n | n n n |
Notes. Numerical verification of Eq. (16) using the TD test sliced with a plane Σ. The column Test labels the different test cases; Σ is the plane interface separating 𝒱a and 𝒱b; H(B, 𝒱a) and H(B, 𝒱b) are the relative magnetic helicities of the subvolumes 𝒱a and 𝒱b, respectively, whereas δH is the nonaddictive term; Hsum and H(B, 𝒱) are the RHS and LHS of Eq. (16), respectively; ϵ is the error in percentage between H(B, 𝒱) and Hsum as defined in Eq. (40); bcAp (respectively bcA) is a triplet representing the integration direction for the computation of the vector potential Ap (respectively A) for the volume 𝒱, 𝒱a, and 𝒱b, respectively; similarly, dVCAp (respectively dVCA) is a triplet representing the boundary condition for the computation of the vector potential Ap (respectively A) on bcAp (respectively bcA), for the volume 𝒱, 𝒱a, and 𝒱b, respectively. See Sect. 5 for additional details.
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