| -ts_theta_theta <Theta> | - Location of stage (0<Theta<=1) | |
| -ts_theta_endpoint <flag> | - Use the endpoint (like Crank-Nicholson) instead of midpoint form of the Theta method | |
| -ts_theta_initial_guess_extrapolate <flg> | - Extrapolate stage initial guess from previous solution (sometimes unstable) |
-ts_type theta -ts_theta_theta 1.0 corresponds to backward Euler (TSBEULER)
-ts_type theta -ts_theta_theta 0.5 corresponds to the implicit midpoint rule
-ts_type theta -ts_theta_theta 0.5 -ts_theta_endpoint corresponds to Crank-Nicholson (TSCN)
This method can be applied to DAE.
This method is cast as a 1-stage implicit Runge-Kutta method.
Theta | Theta
-------------
| 1
For the default Theta=0.5, this is also known as the implicit midpoint rule.
0 | 0 0
1 | 1-Theta Theta
-------------------
| 1-Theta Theta
For the default Theta=0.5, this is the trapezoid rule (also known as Crank-Nicolson, see TSCN).
To apply a diagonally implicit RK method to DAE, the stage formula
Y_i = X + h sum_j a_ij Y'_j
is interpreted as a formula for Y'_i in terms of Y_i and known values (Y'_j, j<i)