petsc-3.5.4 2015-05-23
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TSTHETA

DAE solver using the implicit Theta method

Options Database

-ts_theta_theta <Theta> - Location of stage (0<Theta<=1) -ts_theta_extrapolate <flg> Extrapolate stage solution from previous solution (sometimes unstable) -ts_theta_endpoint <flag> - Use the endpoint (like Crank-Nicholson) instead of midpoint form of the Theta method

Notes

``` -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.

When the endpoint variant is chosen, the method becomes a 2-stage method with first stage explicit

```  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)