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 \(y_i = 0\).

Level#

beginner

Location#

src/ts/impls/rosw/rosw.c

Examples#

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

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