:orphan:
# 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 <type> -*** the class of general linear method (irks)
- ***-ts_gl_rtol <tol>  -*** relative error
- ***-ts_gl_atol <tol>  -*** absolute error
- ***-ts_gl_min_order <p> -*** minimum order method to consider (default=1)
- ***-ts_gl_max_order <p> -*** maximum order method to consider (default=3)
- ***-ts_gl_start_order <p> -*** order of starting method (default=1)
- ***-ts_gl_complete <method> -*** method to use for completing the step (rescale-and-modify or rescale)
- ***-ts_adapt_type <method> -*** 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

```none
  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

```none
  [ 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

```none
  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

```none
  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

```none
  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 (`TSIRKS`).  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

```none
  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

- **** -*** 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.
- **** -*** John Butcher, Numerical methods for ordinary differential equations, second edition, Wiley, 2009.



## See Also
 [](ch_ts), `TSCreate()`, `TS`, `TSSetType()`, `TSType`

## Level
beginner

## Location
<A HREF="PETSC_DOC_OUT_ROOT_PLACEHOLDER/src/ts/impls/implicit/glle/glle.c.html#TSGLLE">src/ts/impls/implicit/glle/glle.c</A>


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[Edit on GitLab](https://gitlab.com/petsc/petsc/-/edit/release/src/ts/impls/implicit/glle/glle.c)


[Index of all TS routines](index.md)  
[Table of Contents for all manual pages](/manualpages/index.md)  
[Index of all manual pages](/manualpages/singleindex.md)