Struct BackwardEuler 

pub struct BackwardEuler {
    pub dt: Scalar,
}

Implicit, single-stage, first-order, fixed-step, Runge-Kutta method.¹

\frac{dy}{dt} = f(t, y)

t_{n+1} = t_n + h

k_1 = f(t_{n+1}, y_{n+1})

y_{n+1} = y_n + hk_1

---

1. Also known as the backward Euler method. ↩

Fields

dt: Scalar

Fixed value for the time step.

Trait Implementations

impl Debug for BackwardEuler

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl Default for BackwardEuler

fn default() -> BackwardEuler

Returns the “default value” for a type.

impl<T> FixedStep<T> for BackwardEuler

fn dt(&self) -> Quantity<T>

Returns the time step.

impl<Y, J, U, V, T> Implicit<Y, J, U, V, T> for BackwardEuler

where Y: Differentiable<T> + Tensor, Derivative<Y, T>: Mul<Quantity<T>, Output = Y>, J: Differentiable<T> + Tensor + TensorArray, Derivative<J, T>: Mul<Quantity<T>, Output = J>, for<'a> &'a Y: Sub<&'a Y, Output = Y>, U: TensorVec<Item = Y>, V: TensorVec<Item = Derivative<Y, T>>,

fn residual( &self, function: impl FnMut(Quantity<T>, &Y) -> Result<Derivative<Y, T>, IntegrationError>, _t: Quantity<T>, y: &Y, t_trial: Quantity<T>, y_trial: &Y, dt: Quantity<T>, ) -> Result<Y, String>

fn hessian( &self, jacobian: impl FnMut(Quantity<T>, &Y) -> Result<Derivative<J, T>, IntegrationError>, _t: Quantity<T>, _y: &Y, t_trial: Quantity<T>, y_trial: &Y, dt: Quantity<T>, ) -> Result<J, String>

fn integrate( &self, function: impl FnMut(Quantity<T>, &Y) -> Result<Derivative<Y, T>, IntegrationError>, jacobian: impl FnMut(Quantity<T>, &Y) -> Result<Derivative<J, T>, IntegrationError>, time: &[Quantity<T>], initial_condition: Y, solver: impl RootFinding<Y, J, Y>, ) -> Result<(Times<T>, U, V), IntegrationError>

Solves an initial value problem by implicitly integrating a system of ordinary differential equations.

impl<X, F, J, UX, UR, T> ImplicitRates<X, F, J, UX, UR, T> for BackwardEuler

where X: Differentiable<T> + Tensor, Derivative<X, T>: Tensor, for<'a, 'a> &'a X: Sub<&'a X, Output = X> + Div<Quantity<T>, Output = Derivative<X, T>>, UX: TensorVec<Item = X>, UR: TensorVec<Item = Derivative<X, T>>,

fn integrate( &self, rate: impl FnMut(Quantity<T>, &X) -> Result<Derivative<X, T>, String>, residual: impl FnMut(Quantity<T>, &X, &Derivative<X, T>) -> Result<F, String>, tangent: impl FnMut(Quantity<T>, &X, &Derivative<X, T>, Quantity<T>) -> Result<J, String>, solver: impl RootFinding<F, J, X>, equality_constraint: impl Fn(Quantity<T>) -> EqualityConstraints, sparse: Option<SparseSolver>, time: &[Quantity<T>], initial_state: X, ) -> Result<(Times<T>, UX, UR), IntegrationError>

Solves an initial value problem for a first-order system, given the residual of the system as a function of time, state, and rate, and its tangent with respect to the state.

impl<Y, U> OdeIntegrator<Y, U> for BackwardEuler

where Y: Tensor, U: TensorVec<Item = Y>,