ISSDE Basic Theory 基本理论

ISSDE Reference

• Application of Collision Simulation Program Based on Stochastic Differential Equations in Plasma
• ​ISSDE program
• ​Fast ion loss in EAST
• ​ICRF-NBI synergy produced fast ion loss

ISSDE Basic Algorithm

• Splitting Method

The equations that ISSDE mainly requires to be solved are ¹
\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}
where \mathbf{a}_{c} is the acceleration caused by the Coulomb collision. According to splitting method, the equation can be regarded as a vector field \boldsymbol{F}(\boldsymbol{x},\boldsymbol{v})
\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]\ ,
Therefore, the above system can be divided into four subsystems and then solved separately. These subsystems are
\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}
\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}
\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}
where
\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]
\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}
The main purpose is that for the deterministic parts, which are \boldsymbol{\phi}_t ^{F_1},\ \boldsymbol{\phi}_t ^{F_2},\ \boldsymbol{\phi}_t ^{F_3}, we can use the volume-preserving algorithm to calculte, while the Coulomb collision part, which is \boldsymbol{\phi}_t ^{F_4}, still uses the Newton-Raphson method or the quasi-Newton method for calculation. In order to construct a second-order algorithm of implicit midpoint similar to Boris algorithm, we can use such a combination
\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}\ .
Then calculate each part separately. When there is no collision, it reduces to the normal Boris algorithm as
\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}\ .

• Volumn-preserving algorithm²
• Quasi-Newton Method

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1. 1. Yifeng Zheng et al. 2021 Chin. Phys. B 30 095201
2. 2. Yang He et al. 2015 Phys. Plasma 22 124503

Physical Process Calculated by ISSDE

• Fast ion trajectory
• Electron-Electron collision
• Electron-Ion collision
• Ion-Ion collision
• Fast ion trajectory in Tokamak
• Fast ion loss under the effect of ripple field, MHD perturbation field, TBM field and collision
• Fast ion loss with LCFS boundary, first wall and limiters
• Slowing down process in Tokamak
• Evolution of fast ion distribution
• Heat load on first wall and limiters³
• Devices can be calculated
  • EAST
  • CFETR
  • BEST
  • CFEDR
  • FRC
  • Stellarator

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1. 3. Yifeng Zheng et al. 2023 Nucl. Fusion 63 046016

ISSDE Interview Setting

• EFIT provides the equilibrium field
• ONETWO provides the temperature and density profiles
• NUBEAM provides the initial distribution of fast ion
• ANSYS provides TBM field
• Analysis solution provide perturbation field
• ANSYS provides ripple field
• Analysis solution provide ripple field
• Analysis solution provide MHD perturbation field
• Engineering data for first wall and limiters
• TORIC provide ICRF field

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