Hello, I’m struggling to understand the mathematical background of the one-step theta scheme implemented in deal.ii adapter linear_elasticity.cc. In there the rhs is implemented as:

RHS=(M - theta * (1-theta) * delta_t^2*K) * V_n - delta_t * K * D_n + delta_t * theta * F_n+1 + delta_t * (1-theta) * F_n

I do not get the original time integration equation which leads to the above equation. I have a look in deal.ii step-23 tutorial which provides an explanation of the Rothe method, which also uses theta parameter and gives identical LHS:

https://www.dealii.org/current/doxygen/deal.II/step_23.html

Following that, the rhs should be:

RHS=M * (D_n+delta_t * V_n) - delta_t^2 * theta * (1-theta) * K * D_n + delta_t^2 * theta * (theta * F_n+1 + (1-theta)*F_n)

Any help to explain the differences would be appreciated.