Implicit steppers#
Header: <pressio/ode_steppers_implicit.hpp>
Public namespace: pressio::ode
Scope#
An “implicit stepper” in pressio is an abstraction that represents “how” to take a step when applying an implicit scheme to initial value problems expressable as
(1)#\[ \frac{d \boldsymbol{y}}{dt} =
\boldsymbol{f}(\boldsymbol{y},t; ...), \qquad y(t_0) = y_0\]
or with a mass matrix:
(2)#\[ M(\boldsymbol{y}, t, ...) \frac{d \boldsymbol{y}}{dt} =
\boldsymbol{f}(\boldsymbol{y},t; ...), \qquad y(t_0) = y_0\]
or directly in fully discrete form as:
(3)#\[ \boldsymbol{R}(\boldsymbol{y_{n}}, \boldsymbol{y_{n-1}}, ..., t_n, dt_n; ...) = \boldsymbol{0}, \qquad y(t_0) = y_0\]
where \(y\) is the state, \(f\) is the right hand side (RHS), \(t\) is the independent variable, \(M\) is the mass matrix, and \(R\) is the residual. Note that both \(f\) and \(M\) potentially depend on the state and \(t\).
Recall the definition of implicit methods
Implicit methods update the state by solving a (potentially nonlinear) system of equations involving the current, the predicted state and possibly previous states.