First-order system

Rewriting as a first-order system in the stresses and velocities $(v_{x}, v_{y}, v_{z}) = \frac{\partial}{\partial t}(u_{x}, u_{y}, u_{z})$ gives

$$\begin{bmatrix}\rho D_{t}&0&0&-D_{x}&0&0&-D_{y}&0&-D_{z}\\ 0&\rho D_{t}&0&0&-D_{y}&0&-D_{x}&-D_{z}&0\\ 0&0&\rho D_{t}&0&0&0&-D_{z}&-D_{y}&-D_{x}\\ -D_{x}&0&0&\beta D_{t}&\gamma D_{t}&\gamma D_{t}&0&0&0\\ 0&-D_{y}&0&\gamma D_{t}&\beta D_{t}&\gamma D_{t}&0&0&0\\ 0&0&-D_{z}&\gamma D_{t}&\gamma D_{t}&\beta D_{t}&0&0&0\\ -D_{y}&-D_{x}&0&0&0&0&\alpha D_{t}&0&0\\ 0&-D_{z}&-D_{y}&0&0&0&0&\alpha D_{t}&0\\ -D_{z}&0&-D_{x}&0&0&0&0&0&\alpha D_{t}\\ \end{bmatrix}\begin{bmatrix}v_{x}\\ v_{y}\\ v_{z}\\ \sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \sigma_{xy}\\ \sigma_{xz}\\ \sigma_{yz}\\ \end{bmatrix}={\bf0}$$

with

$$\alpha = \frac{1}{\mu}\hspace{0.5in}\beta = \frac{\mu+\lambda}{\mu\left(2\mu+3\lambda\right)}\hspace{0.5in}\gamma =\frac{\lambda}{2\mu\left(2\mu+3\lambda\right)}$$

• Symmetric hyperbolic even if $\alpha$ , $\beta$ , $\lambda$ and $\rho$ are spatially varying.
• Coupling of time derivatives (this does not occur in the other systems seen so far). Thus a scalar staggered grid is out of the question (need $\sigma_{xx}$ , $\sigma_{yy}$ and $\sigma_{zz}$ at same grid point, at same time step).

Fix: vectorized differences.