petsc-3.6.0 2015-06-09
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# TSROSW

ODE solver using Rosenbrock-W schemes These methods are intended for problems with well-separated time scales, especially when a slow scale is strongly nonlinear such that it is expensive to solve with a fully implicit method. The user should provide the stiff part of the equation using TSSetIFunction() and the non-stiff part with TSSetRHSFunction().

### Notes

This method currently only works with autonomous ODE and DAE.

Consider trying TSARKIMEX if the stiff part is strongly nonlinear.

### Developer notes

Rosenbrock-W methods are typically specified for autonomous ODE

 udot = f(u)


by the stage equations

 k_i = h f(u_0 + sum_j alpha_ij k_j) + h J sum_j gamma_ij k_j


and step completion formula

 u_1 = u_0 + sum_j b_j k_j


with step size h and coefficients alpha_ij, gamma_ij, and b_i. Implementing the method in this form would require f(u) and the Jacobian J to be available, in addition to the shifted matrix I - h gamma_ii J. Following Hairer and Wanner, we define new variables for the stage equations

 y_i = gamma_ij k_j


The k_j can be recovered because Gamma is invertible. Let C be the lower triangular part of Gamma^{-1} and define

 A = Alpha Gamma^{-1}, bt^T = b^T Gamma^{-1}


to rewrite the method as

 [M/(h gamma_ii) - J] y_i = f(u_0 + sum_j a_ij y_j) + M sum_j (c_ij/h) y_j

 u_1 = u_0 + sum_j bt_j y_j


where we have introduced the mass matrix M. Continue by defining

 ydot_i = 1/(h gamma_ii) y_i - sum_j (c_ij/h) y_j


or, more compactly in tensor notation

 Ydot = 1/h (Gamma^{-1} \otimes I) Y .


Note that Gamma^{-1} is lower triangular. With this definition of Ydot in terms of known quantities and the current stage y_i, the stage equations reduce to performing one Newton step (typically with a lagged Jacobian) on the equation

 g(u_0 + sum_j a_ij y_j + y_i, ydot_i) = 0


with initial guess y_i = 0.