petsc-master 2020-01-17
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# TSGLLE

DAE solver using implicit General Linear methods These methods contain Runge-Kutta and multistep schemes as special cases. These special cases have some fundamental limitations. For example, diagonally implicit Runge-Kutta cannot have stage order greater than 1 which limits their applicability to very stiff systems. Meanwhile, multistep methods cannot be A-stable for order greater than 2 and BDF are not 0-stable for order greater than 6. GL methods can be A- and L-stable with arbitrarily high stage order and reliable error estimates for both 1 and 2 orders higher to facilitate adaptive step sizes and adaptive order schemes. All this is possible while preserving a singly diagonally implicit structure.

### Options database keys

 -ts_gl_type - the class of general linear method (irks) -ts_gl_rtol - relative error -ts_gl_atol - absolute error -ts_gl_min_order

- minimum order method to consider (default=1) -ts_gl_max_order

- maximum order method to consider (default=3) -ts_gl_start_order

- order of starting method (default=1) -ts_gl_complete - method to use for completing the step (rescale-and-modify or rescale) -ts_adapt_type - adaptive controller to use (none step both)

### Notes

This integrator can be applied to DAE.

Diagonally implicit general linear (DIGL) methods are a generalization of diagonally implicit Runge-Kutta (DIRK). They are represented by the tableau

```  A  |  U
-------
B  |  V
```

combined with a vector c of abscissa. "Diagonally implicit" means that A is lower triangular. A step of the general method reads

```  [ Y ] = [A  U] [  Y'   ]
[X^k] = [B  V] [X^{k-1}]
```

where Y is the multivector of stage values, Y' is the multivector of stage derivatives, X^k is the Nordsieck vector of the solution at step k. The Nordsieck vector consists of the first r moments of the solution, given by

```  X = [x_0,x_1,...,x_{r-1}] = [x, h x', h^2 x'', ..., h^{r-1} x^{(r-1)} ]
```

If A is lower triangular, we can solve the stages (Y,Y') sequentially

```  y_i = h sum_{j=0}^{s-1} (a_ij y'_j) + sum_{j=0}^{r-1} u_ij x_j,    i=0,...,{s-1}
```

and then construct the pieces to carry to the next step

```  xx_i = h sum_{j=0}^{s-1} b_ij y'_j  + sum_{j=0}^{r-1} v_ij x_j,    i=0,...,{r-1}
```

Note that when the equations are cast in implicit form, we are using the stage equation to define y'_i in terms of y_i and known stuff (y_j for j<i and x_j for all j).

Error estimation

At present, the most attractive GL methods for stiff problems are singly diagonally implicit schemes which posses Inherent Runge-Kutta Stability (IRKS). These methods have r=s, the number of items passed between steps is equal to the number of stages. The order and stage-order are one less than the number of stages. We use the error estimates in the 2007 paper which provide the following estimates

```  h^{p+1} X^{(p+1)}          = phi_0^T Y' + [0 psi_0^T] Xold
h^{p+2} X^{(p+2)}          = phi_1^T Y' + [0 psi_1^T] Xold
h^{p+2} (dx'/dx) X^{(p+1)} = phi_2^T Y' + [0 psi_2^T] Xold
```

These estimates are accurate to O(h^{p+3}).

Changing the step size

We use the generalized "rescale and modify" scheme, see equation (4.5) of the 2007 paper.

### References

 1 - John Butcher and Z. Jackieweicz and W. Wright, On error propagation in general linear methods for ordinary differential equations, Journal of Complexity, Vol 23, 2007. 2 - John Butcher, Numerical methods for ordinary differential equations, second edition, Wiley, 2009.