Task description
Likely the most invasive step in the attenuation implementation.
For the Runge-Kutta integration, we need strain at the next time step. In specfem3d this is done by taylor expanding the current strain $\partial u_i/\partial x_j$ in time such that
$$\frac{\partial u_i (t+\Delta t)}{\partial x_j} = \frac{\partial u_i (t)}{\partial x_j} + \Delta \frac{\partial \dot{u}_i (t)}{\partial x_j} = \frac{\partial u_i (t)}{\partial x_j} + \Delta \frac{\partial v_i (t)}{\partial x_j}$$
So that we can compute $\epsilon^n$ and $\epsilon^{n+1}$ from the displacement and velocity gradients.
Which are used in the computation of the attenuation memory variable, e.g., for $\kappa$:
$$R^{\ell,n+1}_\kappa = \alpha^\ell \cdot R^{\ell,n}_\kappa + \kappa_{\text{GLL}} \cdot \Psi^\ell_{\kappa,\text{GLL}} \cdot \left(\beta^\ell_{\text{RK}} \cdot S_n + \gamma^\ell_{\text{RK}} \cdot S_{n+1}\right),$$
where, for the computation of $R_\kappa$, $S^{n}$ is the trace of the strain $\epsilon^{n}$.
So this involves the gradient computation of the velocity. With subsequent storage of
Acceptance criteria
