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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 strictly lower triangular part of \(\Gamma^{-1}\) and define

\[ A = \Alpha \Gamma^{-1}, \quad \hat{b}^T = b^T \Gamma^{-1} \]

to rewrite the method as

\[ [M/(h \gamma_{ii}) - J] y_i = f(u_0 + \sum_j a_{ij} y_j) + M \sum_j (c_{ij}/h) y_j \\ u_1 = u_0 + \sum_j \hat{b}_j y_j \]

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