petsc-3.7.7 2017-09-25

TSTHETA

DAE solver using the implicit Theta method

	-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_adapt <flg>	- Use time-step adaptivity with 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)