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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.

Since this uses a single linear solve per time-step if you wish to lag the jacobian or preconditioner computation you must use also -snes_lag_jacobian_persists true or -snes_lag_jacobian_preconditioner true

## 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.

## See Also

TS: Scalable ODE and DAE Solvers, `TSCreate()`, `TS`, `TSSetType()`, `TSRosWSetType()`, `TSRosWRegister()`, `TSROSWTHETA1`, `TSROSWTHETA2`, `TSROSW2M`, `TSROSW2P`, `TSROSWRA3PW`, `TSROSWRA34PW2`, `TSROSWRODAS3`, `TSROSWSANDU3`, `TSROSWASSP3P3S1C`, `TSROSWLASSP3P4S2C`, `TSROSWLLSSP3P4S2C`, `TSROSWGRK4T`, `TSROSWSHAMP4`, `TSROSWVELDD4`, `TSROSW4L`, `TSType`

## Level

beginner

## Location

src/ts/impls/rosw/rosw.c

## Examples

src/ts/tutorials/ex32.c
src/ts/tutorials/ex51.c
src/ts/tutorials/ex41.c
src/ts/tutorials/ex40.c
src/ts/tutorials/ex8.c

Index of all TS routines
Table of Contents for all manual pages
Index of all manual pages