Example PDEs¶
The following PDEs all have the form solvable by PyPDE:
\[\frac{\partial\mathbf{Q}}{\partial t}+\nabla\mathbf{F}\left(\mathbf{Q},\nabla\mathbf{Q}\right)+B\left(\mathbf{Q}\right)\cdot\nabla\mathbf{Q}=\mathbf{S}\left(\mathbf{Q}\right)\]
2D Reactive Euler¶
\[\begin{split}\mathbf{Q}=\left(\begin{array}{c}
\rho\\
\rho E\\
\rho v_{1}\\
\rho v_{2}\\
\rho\lambda
\end{array}\right)\quad\mathbf{F}_{i}=\left(\begin{array}{c}
\rho v_{i}\\
\left(\rho E+p\right)v_{i}\\
\rho v_{i}v_{1}+\delta_{i1}p\\
\rho v_{i}v_{2}+\delta_{i2}p\\
\rho v_{i}\lambda
\end{array}\right)\quad B_{i}=0\quad\mathbf{S}=\left(\begin{array}{c}
0\\
0\\
0\\
0\\
-\rho\lambda K\left(T\right)
\end{array}\right)\end{split}\]
where \(K\) is a (potentially large) function depending on temperature \(T\).
3D Godunov-Romenski¶
\[\begin{split}\mathbf{Q}=\begin{pmatrix}\rho\\
\rho E\\
\rho v_{1}\\
\rho v_{2}\\
\rho v_{3}\\
A_{11}\\
A_{12}\\
A_{13}\\
A_{21}\\
A_{22}\\
A_{23}\\
A_{31}\\
A_{32}\\
A_{33}
\end{pmatrix}\quad\mathbf{F}_{i}=\begin{pmatrix}\rho v_{i}\\
\rho Ev_{i}+\mathbf{\Sigma_{i}}\cdot\mathbf{v}\\
\rho v_{i}v_{1}+\mathbf{\Sigma_{i1}}\\
\rho v_{i}v_{2}+\mathbf{\Sigma_{i2}}\\
\rho v_{i}v_{3}+\mathbf{\Sigma_{i3}}\\
\delta_{i1}\mathbf{A_{1}}\cdot\mathbf{v}\\
\delta_{i2}\mathbf{A_{1}}\cdot\mathbf{v}\\
\delta_{i3}\mathbf{A_{1}}\cdot\mathbf{v}\\
\delta_{i1}\mathbf{A_{2}}\cdot\mathbf{v}\\
\delta_{i2}\mathbf{A_{2}}\cdot\mathbf{v}\\
\delta_{i3}\mathbf{A_{2}}\cdot\mathbf{v}\\
\delta_{i1}\mathbf{A_{3}}\cdot\mathbf{v}\\
\delta_{i2}\mathbf{A_{3}}\cdot\mathbf{v}\\
\delta_{i3}\mathbf{A_{3}}\cdot\mathbf{v}
\end{pmatrix}\quad B_{i}=v_{i}I_{14}-\left(\begin{array}{cccc}
0_{5} & 0_{3} & 0_{3} & 0_{3}\\
0_{3} & \delta_{i1}v_{1}I_{3} & \delta_{i1}v_{2}I_{3} & \delta_{i1}v_{3}I_{3}\\
0_{3} & \delta_{i2}v_{1}I_{3} & \delta_{i2}v_{2}I_{3} & \delta_{i2}v_{3}I_{3}\\
0_{3} & \delta_{i3}v_{1}I_{3} & \delta_{i3}v_{2}I_{3} & \delta_{i3}v_{3}I_{3}
\end{array}\right)\quad\mathbf{S}=-\frac{1}{\theta_{1}}\left(\begin{array}{c}
\mathbf{0_{5}}\\
\mathbf{\frac{\partial E}{\partial A}_{1}}\\
\mathbf{\frac{\partial E}{\partial A}_{2}}\\
\mathbf{\frac{\partial E}{\partial A}_{3}}
\end{array}\right)\end{split}\]
where \(\theta\) is a (potentially very small) function of \(A\), and now:
\[\Sigma=pI+\rho A^{T}\frac{\partial E}{\partial A}\]