Example PDEs

The following PDEs all have the form solvable by PyPDE:

\[\frac{\partial\mathbf{Q}}{\partial t}+\nabla\mathbf{F}\left(\mathbf{Q},\nabla\mathbf{Q}\right)+B\left(\mathbf{Q}\right)\cdot\nabla\mathbf{Q}=\mathbf{S}\left(\mathbf{Q}\right)\]

3D Navier-Stokes

\[\begin{split}\mathbf{Q}=\left(\begin{array}{c} \rho\\ \rho E\\ \rho v_{1}\\ \rho v_{2}\\ \rho v_{3} \end{array}\right)\quad\mathbf{F}_{i}=\left(\begin{array}{c} \rho v_{i}\\ \rho Ev_{i}+\mathbf{\Sigma_{i}}\cdot\mathbf{v}\\ \rho v_{i}v_{1}+\mathbf{\Sigma_{i1}}\\ \rho v_{i}v_{2}+\mathbf{\Sigma_{i2}}\\ \rho v_{i}v_{3}+\mathbf{\Sigma_{i3}} \end{array}\right)\quad B_{i}=0\quad\mathbf{S}=0\end{split}\]

where:

\[\Sigma=pI-\mu\left(\nabla\mathbf{v}+\nabla\mathbf{v}^{T}-\frac{2}{3}tr\left(\nabla\mathbf{v}\right)I\right)\]

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:

\[\Sigma=pI+\rho A^{T}\frac{\partial E}{\partial A}\]