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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 is an IMEX method.

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`.

## Developer Notes

Rosenbrock-W methods are typically specified for autonomous ODE

    $\dot{u} = 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

    $\dot{y}_i = 1/(h \gamma_ii) y_i - \sum_j (c_{ij}/h) y_j

    $or, more compactly in tensor notation

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

    $Note that $\Gamma^{-1}$ is lower triangular. With this definition of $\dot{Y} 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, \dot{y}_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/ex8.c
src/ts/tutorials/ex51.c
src/ts/tutorials/ex32.c
src/ts/tutorials/ex41.c
src/ts/tutorials/ex40.c

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