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 $y_i = 0$.

Level#

beginner

Location#

src/ts/impls/rosw/rosw.c

Examples#

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

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