# TSGLLE#

DAE solver using implicit General Linear methods [BJW07] [But16]

## Options Database Keys#

the class of general linear method (irks)**-ts_gl_type**- relative error**-ts_gl_rtol**- absolute error**-ts_gl_atol**- **-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)**-**method to use for completing the step (rescale-and-modify or rescale)**-ts_gl_complete**- adaptive controller to use (none step both)**-ts_adapt_type**-

## Notes#

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.

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

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

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

and then construct the pieces to carry to the next step

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

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

Changing the step size

Uses the generalized “rescale and modify” scheme, see equation (4.5) of [BJW07].

## References#

J.C. Butcher, Z. Jackiewicz, and W.M. Wright. Error propagation of general linear methods for ordinary differential equations. *Journal of Complexity*, 23(4-6):560–580, 2007. doi:10.1016/j.jco.2007.01.009.

John Charles Butcher. *Numerical methods for ordinary differential equations*. John Wiley & Sons, 2016.

## See Also#

TS: Scalable ODE and DAE Solvers, `TSCreate()`

, `TS`

, `TSSetType()`

, `TSType`

## Level#

beginner

## Location#

src/ts/impls/implicit/glle/glle.c

Index of all TS routines

Table of Contents for all manual pages

Index of all manual pages