| 8 | | [[math(\boldsymbol{F}(\boldsymbol{x},\boldsymbol{v})=\left[\begin{array}{c}\mathbf{v}\\0\end{array}\right]+\left[\begin{array}{c}0\\ \frac{q}{m}\mathbf{E}\end{array}\right]+\left[\begin{array}{c}0\\ \frac{q}{m}\mathbf{v}\times\mathbf{B}\end{array}\right]+\left[\begin{array}{c}0\\ \mathbf{a}_{c}\end{array}\right]\ ,)]] |
| 9 | | |
| | 8 | ISSDE主要求解的方程是\\ |
| | 9 | [[math(\begin{equation}\begin{cases}\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} & =\mathbf{v}\ ,\\ \frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} & =\frac{q}{m}\mathbf{E}+\frac{q}{m}\mathbf{v}\times\mathbf{B}+\mathbf{a}_{c}\ ,\end{cases}\end{equation})]] \\ |
| | 10 | 其中[[math(\mathbf{a}_{c})]]是有库伦碰撞引起的加速度。根据分裂法，这个方程可以看成一个矢量场[[math(\boldsymbol{F}(\boldsymbol{x},\boldsymbol{v}))]] \\ |
| | 11 | [[math(\boldsymbol{F}(\boldsymbol{x},\boldsymbol{v})=\left[\begin{array}{c}\mathbf{v}\\0\end{array}\right]+\left[\begin{array}{c}0\\ \frac{q}{m}\mathbf{E}\end{array}\right]+\left[\begin{array}{c}0\\ \frac{q}{m}\mathbf{v}\times\mathbf{B}\end{array}\right]+\left[\begin{array}{c}0\\ \mathbf{a}_{c}\end{array}\right]\ ,)]]\\ |
| | 12 | 因此上述系统可以分成4个可以独立求解的部分，子系统是\\ |
| | 13 | [[math(\boldsymbol{\phi}_t ^{F_1}:\begin{cases}\boldsymbol{x}(t)=\boldsymbol{x}_0+t\boldsymbol{v}_0,\ \\ \boldsymbol{v}(t)=\boldsymbol{v}_0,\ \end{cases})]] \\ |
| | 14 | [[math(\boldsymbol{\phi}_t ^{F_2}:\begin{cases}\boldsymbol{x}(t)=\boldsymbol{x}_0,\ \\ \boldsymbol{v}(t)=\boldsymbol{v}_0+tq\boldsymbol{E}(\boldsymbol{x}_0)/m,\ \end{cases})]] \\ |
| | 15 | [[math(\boldsymbol{\phi}_t ^{F_3}:\begin{cases}\boldsymbol{x}(t)=\boldsymbol{x}_0+t\boldsymbol{v}_0,\ \\ \boldsymbol{v}(t)=\exp(-t\frac{q}{m}\hat{\boldsymbol{B}}(\boldsymbol{x}_0))\boldsymbol{v}_0,\ \end{cases})]] \\ |
| | 16 | 其中\\ |
| | 17 | [[math(\hat{\boldsymbol{B}}(\boldsymbol{x})=\left[\begin{array}{ccc}0 & -B_{3}(\boldsymbol{x}) & B_{2}(\boldsymbol{x})\\ B_{3}(\boldsymbol{x}) & 0 & -B_{1}(\boldsymbol{x})\\ -B_{2}(\boldsymbol{x}) & B_{1}(\boldsymbol{x}) & 0 \end{array}\right])]] \\ |
| | 18 | [[math(\boldsymbol{\phi}_t ^{F_4}:\begin{cases}\boldsymbol{x}(t)=\boldsymbol{x}_0,\ \\ \int\rm{d}\boldsymbol{v}=\int\boldsymbol{a}_{c}\rm{d}t\ . \end{cases})]] \\ |
| | 19 | 主要目的是对于确定性部分，[[math(\boldsymbol{\phi}_t ^{F_1},\ \boldsymbol{\phi}_t ^{F_2},\ \boldsymbol{\phi}_t ^{F_3})]]，我们可以使用保体积算法进行求解，对于碰撞部分[[math(\boldsymbol{\phi}_t ^{F_4})]]，依旧采用Newton-Raphson方法或者拟牛顿方法进行计算。为了构造一个类似于Boris算法的一个二阶隐式中点算法，我们使用如下组合\\ |
| | 20 | [[math(\boldsymbol{G}_h ^2=\boldsymbol{\phi}_{h/2} ^{F_1}\circ \boldsymbol{\phi}_{h/2} ^{F_2} \circ \boldsymbol{\phi}_{h/2} ^{F_4} \circ \boldsymbol{\phi}_{h} ^{F_3} \circ \boldsymbol{\phi}_{h/2} ^{F_4}\circ \boldsymbol{\phi}_{h/2} ^{F_2} \circ \boldsymbol{\phi}_{h/2} ^{F_1}\ .)]]\\ |
| | 21 | 然后分别计算每一个部分。当没有碰撞时，可以约化到正常的Boris算法\\ |
| | 22 | [[math(\boldsymbol{G}_h ^2=\boldsymbol{\phi}_{h/2} ^{F_1}\circ \boldsymbol{\phi}_{h/2} ^{F_2} \circ \boldsymbol{\phi}_{h} ^{F_3} \circ \boldsymbol{\phi}_{h/2} ^{F_2} \circ \boldsymbol{\phi}_{h/2} ^{F_1}\ .)]]\\ |
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