Verification of Jabobian

After the magnetic coordinates are constructed, we can evaluate the Jacobian $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ by using directly the definition of the Jacobian, i.e.,

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} = \frac{1}{(\nabla \psi \times \nabla \overline{\theta}) \cdot \nabla \phi},$$

which can be further written as

$$\mathcal{J}_{\ensuremath{\operatorname{new}}} = R (R_{\overline{\theta}} Z_{\psi} - R_{\psi} Z_{\overline{\theta}}),$$ (201)

where the partial differential can be evaluated by using numerical differential schemes. The results obtained by this way should agree with results obtained from the analytical form of the Jacobian. This consistency check provide a verification for the correctness of the theory derivation and numerical implementation. In evaluating the Jacobian by using the analytical form, we may need to evaluate $\nabla \psi$, which finally reduces to evaluating $\nabla \Psi$. The value of $\vert \nabla \Psi \vert$ is obtained numerically based on the numerical data of $\Psi$ given in cylindrical coordinate grids. Then the cubic spline interpolating formula is used to obtain the value of $\vert \nabla \Psi \vert$ at desired points. ( $\mathcal{J}_{\ensuremath{\operatorname{new}}}$ calculated by the second method (i.e. using analytic form) is used in the GTAW code; the first methods are also implemented in the code for the benchmark purpose.) In the following sections, for notation ease, the Jacobiban of the constructed coordinate system will be denoted by $\mathcal{J}$, instead of $\mathcal{J}_{\ensuremath{\operatorname{new}}}$.