petsc-3.3-p7 2013-05-11

TSGL

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

  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

John Butcher and Z. Jackieweicz and W. Wright, On error propagation in general linear methods for ordinary differential equations, Journal of Complexity, Vol 23 (4-6), 2007.

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

TSGL

Options database keys

Notes

References

See Also