## 2 Numerical Methods
### Approximated Polynomials and WMLS
\[J=\sum_{j}w_{j}\Big{(}q_{j}^{*}-q_{j}\Big{)}^{2}, \tag{6}\]  
\[\frac{\partial J}{\partial c_{1}}=\frac{\partial J}{\partial c_{2}}=\frac{ \partial J}{\partial c_{3}}=\frac{\partial J}{\partial c_{4}}=\frac{\partial J }{\partial c_{5}}=0, \tag{7}\]  
\[AC=\Big{[}\sum\Delta w_{j}x_{j}\Delta q_{j}\text{ }\sum w_{j}\Delta y \Delta q_{j}\text{ }\sum w_{j}\Delta x_{j}^{2}\Delta q_{j}\text{ }\sum w_{j}\Delta x_{j} \Delta y_{j}\Delta q_{j}\text{ }\sum w_{j}\Delta y_{j}^{2}\Delta q_{j}\Big{]}^{T}, \tag{8}\]  
\[\text{where }A=\left[\begin{array}{ccccc}\sum w_{j}\Delta x_{j}^{2}&\sum w_{j} \Delta x_{j}\Delta y_{j}&\sum w_{j}\Delta x_{j}^{3}&\sum w_{j}\Delta x_{j}^{2} \Delta y_{j}&\sum w_{j}\Delta x_{j}\Delta y_{j}^{2}\\  
a calculation point and that of the surrounding points, and the pressure is sequentially adjusted using the relaxation factor.  
At each arbitrarily arranged calculation point, the convergence calculation of the pressure Poisson equation at time \(n\), as shown in Equation (11), is performed, and the pressure field at this time is obtained. The right-side term is the time change of divergence of the fields of intermediate velocity \(\mathbf{u}\), which is calculated using Equation (12) from the velocity field around calculation point \(i\).  
\[\
abla^{2}p^{n+1}=\frac{\
abla\cdot\mathbf{u}}{\Delta t} \tag{11}\]  
\[\frac{-}{\mathbf{u}-\mathbf{u}^{n}}=-(\mathbf{u}^{n}\cdot\
abla)\mathbf{u}^{ n}+\upsilon\
abla^{2}\mathbf{u}^{n} \tag{12}\]  
Here, the pressure of the surrounding point \(j\) is assumed as follows:  
\[p_{j}^{*}=p_{i}+\frac{\partial p^{*}}{\partial x}\Delta x_{j}+\frac{\partial p ^{*}}{\partial y}\Delta y_{j}+\frac{1}{2}\frac{\partial^{2}p^{*}}{\partial x^ {2}}\Delta x_{j}^{2}+\frac{\partial^{2}p^{*}}{\partial x\partial y}\Delta x_{ j}\Delta y_{j}+\frac{1}{2}\frac{\partial^{2}p^{*}}{\partial y^{2}}\Delta y_{j}^{2}. \tag{13}\]  
The coefficients \(d_{1}\)-\(d_{5}\) are defined as  
\[d_{1}=\frac{\partial p^{*}}{\partial x},\ d_{2}=\frac{\partial p^{*}}{ \partial y},\ d_{3}=\frac{1}{2}\frac{\partial^{2}p^{*}}{\partial x^{2}},\ d_{4}=\frac{\partial^{2}p^{*}}{\partial x \partial y},\ d_{5}=\frac{1}{2}\frac{\partial^{2}p^{*}}{\partial y^{2}}. \tag{14}\]  
An expression similar to Equation (8) is derived based on the condition that the sum of the squares multiplied by the weight of the residual between the assumed pressure and the actual pressure is minimized. Splitting the terms by calculation and surrounding points, it becomes Equation (15). \[AD=\left[ \begin{array}{c}\sum w_{j}\Delta x_{j}\Delta p_{j}\\ \sum w_{j}\Delta y\Delta p_{j}\\ \sum w_{j}\Delta x_{j}^{2}\Delta p_{j}\\ \sum w_{j}\Delta x_{j}\Delta y_{j}\Delta p_{j}\\ \sum w_{j}\Delta y_{j}^{2}\Delta p_{j}\end{array} \right]-\left[ \begin{array}[ ]{c}\sum w_{j}\Delta x_{j}\\ \sum w_{j}\Delta y_{j}\\ \sum w_{j}\Delta x_{j}^{2}\\ \sum w_{j}\Delta x_{j}\Delta y_{j}\\ \sum w_{j}\Delta y_{j}^{2}\end{array} \right]p_{i}, \tag{15}\]  
\[\text{where }D=[d_{1}\ d_{2}\ d_{3}\ d_{4}\ d_{5}]^{\text{T}}. \tag{16}\]  
Equation (15) can be expressed as follows by rearranging \(d_{3}\) and \(d_{5}\), which correspond to the second-order spatial derivatives of the pressure used in the solution of the Poisson equation. \[d_{3}=\frac{1}{2}\frac{\partial^{2}p}{\partial x^{2}}=f_{3}-g_{3}p_{i},\ d_{5}= \frac{1}{2}\frac{\partial^{2}p}{\partial y^{2}}=f_{5}-g_{5}p_{i}. \tag{17}\]  
Here, the coefficients \(f\) and \(g\) can be expressed as in Equation (18). By substituting these expressions into Equation (12) and rearranging for \(p_{i}\), Equation (19) is derived, and this \(p_{i}\) is updated by the relaxation method, as indicated in Equation (20). This updated value is used for the surrounding point when the same calculation is performed at a subsequent calculation point. The update is performed in the same way for all calculation points, and the residual of the pressure \(\varepsilon\) is obtained from Equation (21). \[[f_{1}\ f_{2}\ f_{3}\ f_{4}\ f_{5}]^{T}=A^{T}\left(