Problem Definition

Problem format

Before attempting to solve numerically a problem described by partial differential equation(s), it is necessary to express the equation(s) in a form that can be understood and processed by the solver.

Let us first define 1D discretization grids, for $n \in \mathbb{N}$:

\[a =: x_1 < \cdots < x_n := b \quad \text{ and } \quad t_0 < \cdots < t_{end} \; .\]

The number of points $n$ in the spatial discretization should be chosen based on the desired level of accuracy and the complexity of the solution.

We then need to consider the following system of quasilinear partial differential equations:

\[c(x,t,u,u_x)u_t = x^{-m}(x^m f(x,t,u,u_x))_x + s(x,t,u,u_x)\]

where $m$ designates the symmetry of the problem ($m=0$, $m=1$ or $m=2$ denoting cartesian, polar cylindrical and polar spherical coordinates respectively), while the functions $c(x,t,u,u_x)$, $f(x,t,u,u_x)$ and $s(x,t,u,u_x)$ represent the capacity, flux and source terms respectively.
The capacity term $c(x,t,u,u_x)$ must be expressed as a diagonal matrix (see SkeelBerzins.mass_matrix), this means that the relationship between the partial derivatives with respect to time is limited to being multiplied by a diagonal matrix $c(x,t,u,u_x)$.

Similarly, the boundary conditions associated to the PDE(s) must conform to a specific format, which is outlined as follows:

\[p^i(x,t,u) + q^i(x,t)f^i(x,t,u,u_x) = 0 \quad \text{for } x=a \text{ or } x=b\]

with $i=1,\cdots,$npde (number of PDEs).

The package is based on the spatial discretization method derived in reference [1], which results in a second-order accurate method in space.

Please refer to the section on solvers to understand how to implement the problem in the required format.