|-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)