BackgroundΒΆ
We can solve any system of PDEs of the form:
\[\begin{split}\frac{\partial\mathbf{Q}}{\partial t} & +\frac{\partial}{\partial x_{1}}\mathbf{F}_{1}\left(\mathbf{Q},\frac{\partial\mathbf{Q}}{\partial x_{1}},\ldots,\frac{\partial\mathbf{Q}}{\partial x_{n}}\right)+\cdots+\frac{\partial}{\partial x_{n}}\mathbf{F}_{n}\left(\mathbf{Q},\frac{\partial\mathbf{Q}}{\partial x_{1}},\ldots,\frac{\partial\mathbf{Q}}{\partial x_{n}}\right)\\
& +B_{1}\left(\mathbf{Q}\right)\frac{\partial\mathbf{Q}}{\partial x_{1}}+\cdots+B_{n}\left(\mathbf{Q}\right)\frac{\partial\mathbf{Q}}{\partial x_{n}}\\
& =\mathbf{S}\left(\mathbf{Q}\right)\end{split}\]
or, more succinctly:
\[\frac{\partial\mathbf{Q}}{\partial t}+\nabla\mathbf{F}\left(\mathbf{Q},\nabla\mathbf{Q}\right)+B\left(\mathbf{Q}\right)\cdot\nabla\mathbf{Q}=\mathbf{S}\left(\mathbf{Q}\right)\]
See examples of such systems .
If you give the values of \(\mathbf{Q}\) at time \(t=0\) on a rectangular domain in \(\mathbb{R}^n\), then PyPDE will calculate \(\mathbf{Q}\) on the domain at a later time \(t_f\) that you specify.
The boundary conditions at the edges of the domain can be either transitive or periodic.