Analytical solution for PDE-system's IBVP to validate method of lines

Question

I am in search of initial-boundary value problems which are posed in the form of a system of coupled PDE's (not a single PDE) and for which an analytical solution exists at least for some specific initial-boundary conditions.

In particular I would be interested in a PDE's system of one dynamical ($\partial_r$ on lhs) equation coupled to one constraint ($\partial_t$ on lhs) equation.

The generic character of the question is due to the fact that l have not yet come acrross any PDE's system with the above characteristics. Being more specific would probably make the question overdetermined.

The ultimate purpose of my search is to use the analytical solutions as a numerical test for Mathematica Method of Lines code.

I would appreciate any recommendations.

Answer 1

You could use the "method of manufactured solutions (MMS)", which is nicely described in

https://www.comsol.de/blogs/verify-simulations-with-the-method-of-manufactured-solutions/

Essentially, you take some system of PDE, summarized as $$F[u]=r_F,$$ where $[u]$ includes the partial derivatives of $u$ that are used, and right side $r_B$ which is to be chosen later. Denote the boundary conditions similarly as $$B[u]=r_B.$$ Now select some function $p$ that is in the class of admissible functions and perhaps contains qualities expected in a solution.

Then the test system is $$F[u]=F[p],~~ B[u]=B[p],$$ that is, the right sides are computed as $r_F=F[p]$ and $r_B=B[p]$, and $p$ is obviously the exact solution.

• discussion of MMS in a very simple context, with accordingly simple example https://scicomp.stackexchange.com/q/30562/6839

Answer 2

The simplest may be a 1D linear hyperbolic system of conservation laws $q_t + A q_x = 0$, such as the system of linear acoustics \begin{aligned} p_t + u_0 p_x + K_0 u_x &= 0,\\ u_t + p_x/\rho_0 + u_0 u_x &= 0, \end{aligned} the system of linear elasticity, or the system of electromagnetism. In all cases, $q \in \Bbb R^2$ and $A$ is a matrix with real eigenvalues. Almost every initial-and-boundary-value problem for such systems can be solved via the method of characteristics.