Module integrate 

Integration, ODEs, and DAEs.

Structs

BackwardEuler

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

BogackiShampine

Explicit, three-stage, third-order, variable-step, Runge-Kutta method.

BogackiShampineFixedStep

Explicit, three-stage, third-order, fixed-step, Runge-Kutta method.

BogackiShampineTableau

The Bogacki–Shampine 3(2) tableau.

DormandPrince

Explicit, six-stage, fifth-order, variable-step, Runge-Kutta method.

DormandPrinceFixedStep

Explicit, six-stage, fifth-order, fixed-step, Runge-Kutta method.

DormandPrinceTableau

The Dormand–Prince 5(4) tableau.

Euler

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

Flat

A state in a flat vector space: the increment simply adds.

HermiteSegment

Cubic Hermite dense output over one accepted step, built in the field’s Lie algebra rather than on the state itself.

Heun

Explicit, two-stage, second-order, fixed-step, Runge-Kutta method.

ImplicitMidpoint

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

List

A list of independent copies of one field, e.g. every Gauss point’s plastic state across a mesh; an increment reconstructs entry-wise. Composes with Product for a multi-block mesh (Product<List<Field1>, List<Field2>>).

Midpoint

Explicit, two-stage, second-order, fixed-step, Runge-Kutta method.

Product

A composite of two fields; its state is the matching TensorTuple, and an increment reconstructs component-wise. Nests right for three or more fields.

Ralston

Explicit, two-stage, second-order, fixed-step, Runge-Kutta method.

Trapezoidal

Implicit, two-stage, second-order, fixed-step, Runge-Kutta method.

Unimodular

A state acted on by the matrix exponential, X_{n+1} = exp(increment) X_n, staying on the unimodular group (det = 1) whenever the increment is trace-free. The state maps B → A while its algebra element maps A → A, so F_p (Reference → Intermediate) is Unimodular<Intermediate, Reference>.

Verner8

Explicit, thirteen-stage, eighth-order, variable-step, Runge-Kutta method.

Verner9

Explicit, sixteen-stage, ninth-order, variable-step, Runge-Kutta method.

Verner8FixedStep

Explicit, twelve-stage, eighth-order, fixed-step, Runge-Kutta method.

Verner8Tableau

The Verner 8(7) tableau.

Verner9FixedStep

Explicit, fifteen-stage, ninth-order, fixed-step, Runge-Kutta method.

Verner9Tableau

The Verner 9(8) tableau.

Enums

IntegrationError

Possible errors encountered when integrating.

Traits

ButcherTableau

Butcher tableau for an explicit Runge–Kutta method.

EmbeddedTableau

An embedded explicit Runge–Kutta pair.

Explicit

Explicit integrators for ordinary differential equations.

ExplicitDaeFirstOrderMinimize

Integrators for explicit differential-algebraic equations using first-order minimization.

ExplicitDaeFirstOrderRoot

Integrators for explicit differential-algebraic equations using first-order root-finding.

ExplicitDaeSecondOrderMinimize

Integrators for explicit differential-algebraic equations using second-order minimization.

ExplicitDaeVariableStepExplicit

Variable-step explicit integrators for explicit differential-algebraic equations.

ExplicitDaeVariableStepExplicitFirstOrderMinimize

Variable-step explicit integrators for explicit differential-algebraic equations using first-order minimization.

ExplicitDaeVariableStepExplicitFirstOrderRoot

Variable-step explicit integrators for explicit differential-algebraic equations using first-order root-finding.

ExplicitDaeVariableStepExplicitSecondOrderMinimize

Variable-step explicit integrators for explicit differential-algebraic equations using second-order minimization.

ExplicitDaeVariableStepExplicitZerothOrderRoot

Variable-step explicit integrators for explicit differential-algebraic equations using zeroth-order root-finding.

ExplicitDaeVariableStepFirstSameAsLast

First-same-as-last property for explicit differential-algebraic equation integrators.

ExplicitDaeZerothOrderRoot

Integrators for explicit differential-algebraic equations using zeroth-order root-finding.

FixedStep

Fixed-step integrators for ordinary differential equations.

FixedStepExplicit

Fixed-step explicit integrators for ordinary differential equations.

FreeInterpolant

Free (dense-output) interpolant for explicit ordinary differential equation integrators.

ImplicitDaeFirstOrderMinimize

Integrators for implicit differential-algebraic equations using first-order minimization.

ImplicitDaeFirstOrderRoot

Integrators for implicit differential-algebraic equations using first-order root-finding.

ImplicitDaeSecondOrderMinimize

Integrators for implicit differential-algebraic equations using second-order minimization.

ImplicitDaeVariableStepExplicit

Variable-step explicit integrators for implicit differential-algebraic equations.

ImplicitDaeVariableStepExplicitFirstOrderMinimize

Variable-step explicit integrators for implicit differential-algebraic equations using first-order minimization.

ImplicitDaeVariableStepExplicitFirstOrderRoot

Variable-step explicit integrators for implicit differential-algebraic equations using first-order root-finding.

ImplicitDaeVariableStepExplicitSecondOrderMinimize

Variable-step explicit integrators for implicit differential-algebraic equations using second-order minimization.

ImplicitDaeVariableStepExplicitZerothOrderRoot

Variable-step explicit integrators for implicit differential-algebraic equations using zeroth-order root-finding.

ImplicitDaeZerothOrderRoot

Integrators for implicit differential-algebraic equations using zeroth-order root-finding.

ImplicitFirstOrder

Implicit integrators for ordinary differential equations using first-order root-finding.

ImplicitZerothOrder

Implicit integrators for ordinary differential equations using zeroth-order root-finding.

Integrable

The geometry of one integrated state field: how an increment advances the state.

OdeIntegrator

Integrators for ordinary differential equations.

StateEvolution

A model whose internal state evolves as a product of Lie-algebra rates, ready for the field drivers. Self::Drive is the externally-imposed input the rate needs beside the state (e.g. the total deformation gradient).

VariableStep

Variable-step integrators for ordinary differential equations.

VariableStepExplicit

Variable-step explicit integrators for ordinary differential equations.

VariableStepExplicitFirstSameAsLast

First-same-as-last property for explicit ordinary differential equation integrators.

Functions

integrate_euler

Explicit Euler for a single Integrable, one step per interval of time.

integrate_rkmk

Runge–Kutta–Munthe-Kaas: a fixed-step ButcherTableau run in the field’s Lie algebra, with the Integrable::dexpinv correction per stage and a single Integrable::reconstruct per step. Reduces to the plain tableau on a flat field. See super::rkmk_step for the allocation-free single step.

integrate_rkmk_adaptive

Adaptive RKMK: integrate_rkmk with embedded local-error control from the tableau’s D weights. The step is grown or shrunk by 0.9 (tol / e)^{1/p} (clamped to [DT_CUT, 5]), and a step whose error e exceeds abs_tol + rel_tol ‖x_{n+1}‖ is rejected. A rate-evaluation failure is retried with dt *= DT_CUT, the same as a rejected accuracy estimate.

integrate_rkmk_dae_adaptive

Adaptive super::rkmk_dae_step: embedded local-error control from the tableau’s D weights over the span [time[0], time[last]], with the same controller as integrate_rkmk_adaptive. A rejected step costs no endpoint constraint solve. Returns the accepted times, the state history, and the matching algebraic history.

integrate_rkmk_dae_adaptive_first_order_root

integrate_rkmk_dae_adaptive with the algebraic unknown resolved by first-order root-finding at every stage abscissa, built from function/jacobian/solver the same way super::rkmk_dae_step_first_order_root builds it for a single step.

integrate_rkmk_dae_adaptive_second_order_minimize

integrate_rkmk_dae_adaptive with the algebraic unknown resolved by second-order minimization at every stage abscissa, built from function/jacobian/hessian/solver the same way super::rkmk_dae_step_second_order_minimize builds it for a single step.

integrate_rkmk_state

Runs super::integrate_rkmk over a StateEvolution model, sampling drive at each stage time and starting from the model’s own initial state.

integrate_rkmk_state_adaptive

Runs super::integrate_rkmk_adaptive over a StateEvolution model, sampling drive at each stage time and starting from the model’s own initial state. time supplies only the span [time[0], time[last]]; the returned times are the steps the controller accepted.

interpolate_hermite

Finds the containing segment and evaluates it there, over a whole grid of requested times.

rkmk_dae_step

One RKMK step of a semi-explicit DAE: the differential field advances on its manifold while the algebraic unknown is re-solved from its constraint at every stage abscissa.

rkmk_dae_step_first_order_root

rkmk_dae_step with the algebraic unknown resolved by first-order root-finding at every stage abscissa, built from function/jacobian/ solver exactly as ExplicitDaeVariableStepExplicitFirstOrderRoot builds its solution closure for the legacy flat DAE solver — the split between root-finding and minimization is orthogonal to which field the state lives on, so this is the one place that wrapping happens for the RKMK-DAE path. Any super::StateEvolution model that also supplies a residual and its Jacobian in terms of the whole field state gets the manifold-aware stage-equilibrium step for free, without hand-rolling this closure itself.

rkmk_dae_step_second_order_minimize

rkmk_dae_step with the algebraic unknown resolved by second-order minimization at every stage abscissa, built from function/jacobian/ hessian/solver the same way rkmk_dae_step_first_order_root builds it for root-finding — the two are siblings so a model whose equilibrium is naturally posed as a potential (rather than a residual) gets the same manifold-aware stage-equilibrium step.

rkmk_step

Advances point one RKMK step from t to t + dt with the Tab tableau — super::integrate_rkmk without the history, and with the stage-slope buffer scratch passed in so a stepping loop allocates nothing per step. scratch may start empty; its contents are overwritten.

Type Aliases

EvolvedIncrement

The Increment (Lie-algebra) type of a StateEvolution model’s field.

EvolvedState

The Point type of a StateEvolution model’s field.

Ode1

Alias for Euler.

Ode2

Alias for Heun.

Ode3

Alias for BogackiShampineFixedStep.

Ode5

Alias for DormandPrinceFixedStep.

Ode8

Alias for Verner8FixedStep.

Ode9

Alias for Verner9FixedStep.

Ode1be

Alias for BackwardEuler.

Ode23

Alias for BogackiShampine.

Ode45

Alias for DormandPrince.

Ode78

Alias for Verner8.

Ode89

Alias for Verner9.

Times

The variable of integration values.