1.2.11. Numerical Methods

1.2.11.1. Finite Difference Form

Suppose we are going to solve the problem

\[i(\partial_t +\mathbf v \cdot \boldsymbol \nabla) \rho = [ \mathbf H, \rho ],\]

where

\[\begin{split}\rho = \frac{1}{2} \begin{pmatrix} a & b_r + i b_i \\ b_r - i b_i & -a \end{pmatrix},\end{split}\]

and

\[\begin{split}\mathbf H = \frac{1}{2}\begin{pmatrix} h & h'_r + i h'_i\\ h'_r - i h'_i & -h \end{pmatrix}.\end{split}\]

To solve it numerically we need to write down the real equation

\[\begin{split}(\partial_t + \mathbf v \cdot \boldsymbol\nabla) a &= - h'_r b_i + h'_i b_r \\ (\partial_t + \mathbf v \cdot \boldsymbol\nabla) b_r &= - h'_i a + h b_i \\ (\partial_t + \mathbf v \cdot \boldsymbol\nabla) b_i &= - h b_r + h'_r a.\end{split}\]

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