-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) |

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.

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. |

Index of all TS routines

Table of Contents for all manual pages

Index of all manual pages