# Nature of a large systems of first order coupled PDEs

My actual equations are the dynamic equations of a Cosserat rod, which is of the form

$$\frac{\partial{U}} {\partial{t}}+B\frac{\partial{U}} {\partial{s}}+c=0$$

where $$t$$ denotes time, $$s$$ denotes the spatial variable and $$U$$ is an $$18 \times 1$$ vector of dependent variables and $$B$$ is a constant $$18 \times 18$$ matrix given by

$$B=\left[\matrix{O & O & O & O & O & O\\ O & O & O & O & O & O\\O & O & O & O & I & O\\O & O & O & O & O & I\\O & O & P & O & O & O\\O & O & O & Q & O & O}\right]$$

where

\begin{align} O&=\left[\matrix {0 &0 &0\\0 &0 &0\\0 &0 &0}\right], \quad P=10^{7}\left[\matrix {1.02 &0 &0\\0 &1.02 &0\\0 &0 &2.65}\right] \\ I&=\left[\matrix {1 &0 &0\\0 &1 &0\\0 &0 &1}\right], \quad Q=10^{7}\left[\matrix {2.65 &0 &0\\0 &2.65 &0\\0 &0 &1.02}\right] \end{align}

and $$c$$ is a vector of nonlinear functions of the unknowns $$U$$.

The first two rows in $$B$$ are completely zeros, which means the first six equations do not have any spatial derivatives and so are ODEs. I would like to know the nature of the system of equations.

What I have done

I have found the eigenvalues of the matrix $$B$$ (using MATLAB)

\begin{align} &[3194.9, -3194.9, 5151.5, -5151.5, 5151.5, -5151.5, 3194.9, \\ &-3194.9, 5151.5, -5151.5, 3194.9, -3194.9, 0, 0, 0, 0, 0, 0] \end{align}

All the eigenvalues are real, but not distinct (4 eigenvalues have a multiplicity of 3 each, and one eigenvalue has a multiplicity of 6).

Some text books say that if the matrix $$B$$ is diagonalizable, then the system will be hyperbolic, and if all the eigenvalues are real and distinct, the system is strictly hyperbolic.

Using MATLAB, I computed

$$[V,D]= \text{eig}(B)$$

and MATLAB says $$V$$ is of full rank but the determinant is very low ($$-1.4 \cdot 10^{-20}$$). Can I say that my system is hyperbolic?

If yes, which numerical method is suitable for this system? Can I use any explicit finite difference scheme?

• If it wasn't for $c$, this would be easily solvable. If $c$ were independent of $U$, it would be reasonably solvable. But you say $c$ is a non-linear function of $U$. That makes talking about solutions much harder. I think you'll need to give more information about $c$ before anyone can discuss this in a useful manner. Aug 17, 2020 at 3:18

It is definitely hyperbolic since eigenvalues of $$B$$ are real, but it is not strictly hyperbolic. Such an hyperbolic system of balance laws can be solved numerically by using operator splitting (Godunov splitting, Strang splitting, etc.), which consists in alternating between the integration of $$\partial_t U + B\, \partial_s U = 0 \tag{A}$$ and $$\partial_t U + c = 0 \tag{B}$$ The linear system of conservation laws (A) can be solved using e.g the Lax-Friedrichs or Lax-Wendroff schemes. The evolution equation (B) can be solved using the Euler method or other Runge-Kutta schemes for instance.