Changes between Version 5 and Version 6 of theory_Eng

Timestamp:

Feb 17, 2025, 7:27:12 AM (19 months ago)

Author:

Yifeng Zheng

Comment:

--

theory_Eng

@@ -6 +6 @@
 == ISSDE Basic Algorithm
 * Splitting Method
-[[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]\ ,)]]
+The equations that ISSDE mainly requires to be solved are \\
+[[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})]] \\
+where [[math(\mathbf{a}_{c})]] is the acceleration caused by the Coulomb collision. According to splitting method, the equation can be regarded as a vector field [[math(\boldsymbol{F}(\boldsymbol{x},\boldsymbol{v}))]] \\
+[[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]\ ,)]]\\
+Therefore, the above system can be divided into four subsystems and then solved separately. These subsystems are\\
+[[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})]] \\
+[[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})]] \\
+[[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})]] \\
+where\\
+[[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])]] \\
+[[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})]] \\
+The main purpose is that for the deterministic parts, which are [[math(\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 [[math(\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\\
+[[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}\ .)]]\\
+The calculate each part separately. When there is no collision, it reduces to the normal Boris algorithm as\\
+[[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}\ .)]]\\
 
 * Volumn-preserving algorithm