# 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#

## Examples#

src/ts/tutorials/ex40.c

src/ts/tutorials/ex51.c

src/ts/tutorials/ex41.c

src/ts/tutorials/ex8.c

src/ts/tutorials/ex32.c

Index of all TS routines

Table of Contents for all manual pages

Index of all manual pages