Drake
ImplicitEulerIntegrator< T > Class Template Referencefinal

Detailed Description

template<class T>
class drake::systems::ImplicitEulerIntegrator< T >

A first-order, fully implicit integrator with second order error estimation.

This integrator uses the following update rule:

x(t+h) = x(t) + h f(t+h,x(t+h))

where x are the state variables, h is the integration step size, and f() returns the time derivatives of the state variables. Contrast this update rule to that of an explicit first-order integrator:

x(t+h) = x(t) + h f(t, x(t))

Thus implicit first-order integration must solve a nonlinear system of equations to determine both the state at t+h and the time derivatives of that state at that time. Cast as a nonlinear system of equations, we seek the solution to:

x(t+h) − x(t) − h f(t+h,x(t+h)) = 0

given unknowns x(t+h).

This "implicit Euler" method is known to be L-Stable, meaning both that applying it at a fixed integration step to the "test" equation y(t) = eᵏᵗ yields zero (for k < 0 and t → ∞) and that it is also A-Stable. A-Stability, in turn, means that the method can integrate the linear constant coefficient system dx/dt = Ax at any step size without the solution becoming unstable (growing without bound). The practical effect of L-Stability is that the integrator tends to be stable for any given step size on an arbitrary system of ordinary differential equations. See [Lambert, 1991], Ch. 6 for an approachable discussion on stiff differential equations and L- and A-Stability.

This implementation uses Newton-Raphson (NR) and relies upon the obvious convergence to a solution for g = 0 where g(x(t+h)) ≡ x(t+h) − x(t) − h f(t+h,x(t+h)) as h becomes sufficiently small. General implementational details for the Newton method were gleaned from Section IV.8 in [Hairer, 1996].

Error Estimation

In this integrator, we simultaneously take a large step at the requested step size of h as well as two half-sized steps each with step size h/2. The result from two half-sized steps is propagated as the solution, while the difference between the two results is used as the error estimate for the propagated solution. This error estimate is accurate to the second order.

To be precise, let x̅ⁿ⁺¹ be the computed solution from a large step, x̃ⁿ⁺¹ be the computed solution from two small steps, and xⁿ⁺¹ be the true solution. Since the integrator propagates x̃ⁿ⁺¹ as its solution, we denote the true error vector as ε = x̃ⁿ⁺¹ − xⁿ⁺¹. ImplicitEulerIntegrator uses ε* = x̅ⁿ⁺¹ − x̃ⁿ⁺¹, the difference between the two solutions, as the second-order error estimate, because for a smooth system, ‖ε*‖ = O(h²), and ‖ε − ε*‖ = O(h³). See the notes in get_error_estimate_order() for a detailed derivation of the error estimate's truncation error.

In this implementation, ImplicitEulerIntegrator attempts the large full-sized step before attempting the two small half-sized steps, because the large step is more likely to fail to converge, and if it is performed first, convergence failures are detected early, avoiding the unnecessary effort of computing potentially-successful small steps.

Optionally, ImplicitEulerIntegrator can instead use the implicit trapezoid method for error estimation. However, in our testing the step doubling method substantially outperforms the implicit trapezoid method.

  • [Hairer, 1996] E. Hairer and G. Wanner. Solving Ordinary Differential Equations II (Stiff and Differential-Algebraic Problems). Springer, 1996, Section IV.8, p. 118–130.
  • [Lambert, 1991] J. D. Lambert. Numerical Methods for Ordinary Differential Equations. John Wiley & Sons, 1991.
Note
In the statistics reported by IntegratorBase, all statistics that deal with the number of steps or the step sizes will track the large full-sized steps. This is because the large full-sized h is the smallest irrevocable time-increment advanced by this integrator: if, for example, the second small half-sized step fails, this integrator revokes to the state before the first small step. This behavior is similar to other integrators with multi-stage evaluation: the step-counting statistics treat a "step" as the combination of all the stages.
Furthermore, because the small half-sized steps are propagated as the solution, the large full-sized step is the error estimator, and the error estimation statistics track the effort during the large full-sized step. If the integrator is not in full-Newton mode (see ImplicitIntegrator::set_use_full_newton()), most of the work incurred by constructing and factorizing matrices and by failing Newton-Raphson iterations will be counted toward the error estimation statistics, because the large step is performed first.
This integrator uses the integrator accuracy setting, even when run in fixed-step mode, to limit the error in the underlying Newton-Raphson process. See IntegratorBase::set_target_accuracy() for more info.
See also
ImplicitIntegrator class documentation for information about implicit integration methods in general.
Template Parameters
TThe scalar type, which must be one of the default nonsymbolic scalars.

#include <drake/systems/analysis/implicit_euler_integrator.h>

Public Member Functions

 ~ImplicitEulerIntegrator () override=default
 
 ImplicitEulerIntegrator (const System< T > &system, Context< T > *context=nullptr)
 
bool supports_error_estimation () const final
 Returns true, because this integrator supports error estimation. More...
 
int get_error_estimate_order () const final
 Returns the asymptotic order of the difference between the large and small steps (from which the error estimate is computed), which is 2. More...
 
void set_use_implicit_trapezoid_error_estimation (bool flag)
 Set this to true to use implicit trapezoid for error estimation; otherwise this integrator will use step doubling for error estimation. More...
 
bool get_use_implicit_trapezoid_error_estimation ()
 Returns true if the integrator will use implicit trapezoid for error estimation; otherwise it indicates the integrator will use step doubling for error estimation. More...
 
Does not allow copy, move, or assignment
 ImplicitEulerIntegrator (const ImplicitEulerIntegrator &)=delete
 
ImplicitEulerIntegratoroperator= (const ImplicitEulerIntegrator &)=delete
 
 ImplicitEulerIntegrator (ImplicitEulerIntegrator &&)=delete
 
ImplicitEulerIntegratoroperator= (ImplicitEulerIntegrator &&)=delete
 
- Public Member Functions inherited from ImplicitIntegrator< T >
virtual ~ImplicitIntegrator ()
 
 ImplicitIntegrator (const System< T > &system, Context< T > *context=nullptr)
 
int max_newton_raphson_iterations () const
 The maximum number of Newton-Raphson iterations to take before the Newton-Raphson process decides that convergence will not be attained. More...
 
void set_reuse (bool reuse)
 Sets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations (default is true). More...
 
bool get_reuse () const
 Gets whether the integrator attempts to reuse Jacobian matrices and iteration matrix factorizations. More...
 
void set_use_full_newton (bool flag)
 Sets whether the method operates in "full Newton" mode, in which case Jacobian and iteration matrices are freshly computed on every Newton-Raphson iteration. More...
 
bool get_use_full_newton () const
 Gets whether this method is operating in "full Newton" mode. More...
 
void set_jacobian_computation_scheme (JacobianComputationScheme scheme)
 Sets the Jacobian computation scheme. More...
 
JacobianComputationScheme get_jacobian_computation_scheme () const
 
int64_t get_num_derivative_evaluations_for_jacobian () const
 Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for computing the Jacobian matrices since the last call to ResetStatistics(). More...
 
int64_t get_num_jacobian_evaluations () const
 Gets the number of Jacobian computations (i.e., the number of times that the Jacobian matrix was reformed) since the last call to ResetStatistics(). More...
 
int64_t get_num_newton_raphson_iterations () const
 Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process since the last call to ResetStatistics(). More...
 
int64_t get_num_iteration_matrix_factorizations () const
 Gets the number of factorizations of the iteration matrix since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_derivative_evaluations () const
 Gets the number of ODE function evaluations (calls to EvalTimeDerivatives()) used only for the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_derivative_evaluations_for_jacobian () const
 
int64_t get_num_error_estimator_newton_raphson_iterations () const
 Gets the number of iterations used in the Newton-Raphson nonlinear systems of equation solving process for the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_jacobian_evaluations () const
 Gets the number of Jacobian matrix computations used only during the error estimation process since the last call to ResetStatistics(). More...
 
int64_t get_num_error_estimator_iteration_matrix_factorizations () const
 Gets the number of factorizations of the iteration matrix used only during the error estimation process since the last call to ResetStatistics(). More...
 
- Public Member Functions inherited from IntegratorBase< T >
 IntegratorBase (const System< T > &system, Context< T > *context=nullptr)
 Maintains references to the system being integrated and the context used to specify the initial conditions for that system (if any). More...
 
virtual ~IntegratorBase ()=default
 
void Reset ()
 Resets the integrator to initial values, i.e., default construction values. More...
 
void Initialize ()
 An integrator must be initialized before being used. More...
 
StepResult IntegrateNoFurtherThanTime (const T &publish_time, const T &update_time, const T &boundary_time)
 (Internal use only) Integrates the system forward in time by a single step with step size subject to integration error tolerances (assuming that the integrator supports error estimation). More...
 
void IntegrateWithMultipleStepsToTime (const T &t_final)
 Stepping function for integrators operating outside of Simulator that advances the continuous state exactly to t_final. More...
 
bool IntegrateWithSingleFixedStepToTime (const T &t_target)
 Stepping function for integrators operating outside of Simulator that advances the continuous state using a single step to t_target. More...
 
const Context< T > & get_context () const
 Returns a const reference to the internally-maintained Context holding the most recent state in the trajectory. More...
 
Context< T > * get_mutable_context ()
 Returns a mutable pointer to the internally-maintained Context holding the most recent state in the trajectory. More...
 
void reset_context (Context< T > *context)
 Replace the pointer to the internally-maintained Context with a different one. More...
 
const System< T > & get_system () const
 Gets a constant reference to the system that is being integrated (and was provided to the constructor of the integrator). More...
 
bool is_initialized () const
 Indicates whether the integrator has been initialized. More...
 
const T & get_previous_integration_step_size () const
 Gets the size of the last (previous) integration step. More...
 
 IntegratorBase (const IntegratorBase &)=delete
 
IntegratorBaseoperator= (const IntegratorBase &)=delete
 
 IntegratorBase (IntegratorBase &&)=delete
 
IntegratorBaseoperator= (IntegratorBase &&)=delete
 
void set_target_accuracy (double accuracy)
 Request that the integrator attempt to achieve a particular accuracy for the continuous portions of the simulation. More...
 
double get_target_accuracy () const
 Gets the target accuracy. More...
 
double get_accuracy_in_use () const
 Gets the accuracy in use by the integrator. More...
 
const ContinuousState< T > * get_error_estimate () const
 Gets the error estimate (used only for integrators that support error estimation). More...
 
const T & get_ideal_next_step_size () const
 Return the step size the integrator would like to take next, based primarily on the integrator's accuracy prediction. More...
 
void set_fixed_step_mode (bool flag)
 Sets an integrator with error control to fixed step mode. More...
 
bool get_fixed_step_mode () const
 Gets whether an integrator is running in fixed step mode. More...
 
const Eigen::VectorXd & get_generalized_state_weight_vector () const
 Gets the weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...
 
Eigen::VectorBlock< Eigen::VectorXd > get_mutable_generalized_state_weight_vector ()
 Gets a mutable weighting vector (equivalent to a diagonal matrix) applied to weighting both generalized coordinate and velocity state variable errors, as described in the group documentation. More...
 
const Eigen::VectorXd & get_misc_state_weight_vector () const
 Gets the weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More...
 
Eigen::VectorBlock< Eigen::VectorXd > get_mutable_misc_state_weight_vector ()
 Gets a mutable weighting vector (equivalent to a diagonal matrix) for weighting errors in miscellaneous continuous state variables z. More...
 
void request_initial_step_size_target (const T &step_size)
 Request that the first attempted integration step have a particular size. More...
 
const T & get_initial_step_size_target () const
 Gets the target size of the first integration step. More...
 
void set_maximum_step_size (const T &max_step_size)
 Sets the maximum step size that may be taken by this integrator. More...
 
const T & get_maximum_step_size () const
 Gets the maximum step size that may be taken by this integrator. More...
 
double get_stretch_factor () const
 Gets the stretch factor (> 1), which is multiplied by the maximum (typically user-designated) integration step size to obtain the amount that the integrator is able to stretch the maximum time step toward hitting an upcoming publish or update event in IntegrateNoFurtherThanTime(). More...
 
void set_requested_minimum_step_size (const T &min_step_size)
 Sets the requested minimum step size h_min that may be taken by this integrator. More...
 
const T & get_requested_minimum_step_size () const
 Gets the requested minimum step size h_min for this integrator. More...
 
void set_throw_on_minimum_step_size_violation (bool throws)
 Sets whether the integrator should throw a std::exception when the integrator's step size selection algorithm determines that it must take a step smaller than the minimum step size (for, e.g., purposes of error control). More...
 
bool get_throw_on_minimum_step_size_violation () const
 Reports the current setting of the throw_on_minimum_step_size_violation flag. More...
 
get_working_minimum_step_size () const
 Gets the current value of the working minimum step size h_work(t) for this integrator, which may vary with the current time t as stored in the integrator's context. More...
 
void ResetStatistics ()
 Forget accumulated statistics. More...
 
int64_t get_num_substep_failures () const
 Gets the number of failed sub-steps (implying one or more step size reductions was required to permit solving the necessary nonlinear system of equations). More...
 
int64_t get_num_step_shrinkages_from_substep_failures () const
 Gets the number of step size shrinkages due to sub-step failures (e.g., integrator convergence failures) since the last call to ResetStatistics() or Initialize(). More...
 
int64_t get_num_step_shrinkages_from_error_control () const
 Gets the number of step size shrinkages due to failure to meet targeted error tolerances, since the last call to ResetStatistics or Initialize(). More...
 
int64_t get_num_derivative_evaluations () const
 Returns the number of ODE function evaluations (calls to CalcTimeDerivatives()) since the last call to ResetStatistics() or Initialize(). More...
 
const T & get_actual_initial_step_size_taken () const
 The actual size of the successful first step. More...
 
const T & get_smallest_adapted_step_size_taken () const
 The size of the smallest step taken as the result of a controlled integration step adjustment since the last Initialize() or ResetStatistics() call. More...
 
const T & get_largest_step_size_taken () const
 The size of the largest step taken since the last Initialize() or ResetStatistics() call. More...
 
int64_t get_num_steps_taken () const
 The number of integration steps taken since the last Initialize() or ResetStatistics() call. More...
 
void add_derivative_evaluations (double evals)
 Manually increments the statistic for the number of ODE evaluations. More...
 
void StartDenseIntegration ()
 Starts dense integration, allocating a new dense output for this integrator to use. More...
 
const trajectories::PiecewisePolynomial< T > * get_dense_output () const
 Returns a const pointer to the integrator's current PiecewisePolynomial instance, holding a representation of the continuous state trajectory since the last StartDenseIntegration() call. More...
 
std::unique_ptr< trajectories::PiecewisePolynomial< T > > StopDenseIntegration ()
 Stops dense integration, yielding ownership of the current dense output to the caller. More...
 

Additional Inherited Members

- Public Types inherited from ImplicitIntegrator< T >
enum  JacobianComputationScheme { kForwardDifference, kCentralDifference, kAutomatic }
 
- Public Types inherited from IntegratorBase< T >
enum  StepResult {
  kReachedPublishTime = 1, kReachedZeroCrossing = 2, kReachedUpdateTime = 3, kTimeHasAdvanced = 4,
  kReachedBoundaryTime = 5, kReachedStepLimit = 6
}
 Status returned by IntegrateNoFurtherThanTime(). More...
 
- Protected Types inherited from ImplicitIntegrator< T >
enum  ConvergenceStatus { kDiverged, kConverged, kNotConverged }
 
- Protected Member Functions inherited from ImplicitIntegrator< T >
virtual int do_max_newton_raphson_iterations () const
 Derived classes can override this method to change the number of Newton-Raphson iterations (10 by default) to take before the Newton-Raphson process decides that convergence will not be attained. More...
 
bool MaybeFreshenMatrices (const T &t, const VectorX< T > &xt, const T &h, int trial, const std::function< void(const MatrixX< T > &J, const T &h, typename ImplicitIntegrator< T >::IterationMatrix *)> &compute_and_factor_iteration_matrix, typename ImplicitIntegrator< T >::IterationMatrix *iteration_matrix)
 Computes necessary matrices (Jacobian and iteration matrix) for Newton-Raphson (NR) iterations, as necessary. More...
 
void FreshenMatricesIfFullNewton (const T &t, const VectorX< T > &xt, const T &h, const std::function< void(const MatrixX< T > &J, const T &h, typename ImplicitIntegrator< T >::IterationMatrix *)> &compute_and_factor_iteration_matrix, typename ImplicitIntegrator< T >::IterationMatrix *iteration_matrix)
 Computes necessary matrices (Jacobian and iteration matrix) for full Newton-Raphson (NR) iterations, if full Newton-Raphson method is activated (if it's not activated, this method is a no-op). More...
 
bool IsUpdateZero (const VectorX< T > &xc, const VectorX< T > &dxc, double eps=-1.0) const
 Checks whether a proposed update is effectively zero, indicating that the Newton-Raphson process converged. More...
 
ConvergenceStatus CheckNewtonConvergence (int iteration, const VectorX< T > &xtplus, const VectorX< T > &dx, const T &dx_norm, const T &last_dx_norm) const
 Checks a Newton-Raphson iteration process for convergence. More...
 
virtual void DoImplicitIntegratorReset ()
 Derived classes can override this method to perform routines when Reset() is called. More...
 
bool IsBadJacobian (const MatrixX< T > &J) const
 Checks to see whether a Jacobian matrix is "bad" (has any NaN or Inf values) and needs to be recomputed. More...
 
MatrixX< T > & get_mutable_jacobian ()
 
void DoResetStatistics () override
 Resets any statistics particular to a specific integrator. More...
 
void DoReset () final
 Derived classes can override this method to perform routines when Reset() is called. More...
 
const MatrixX< T > & CalcJacobian (const T &t, const VectorX< T > &x)
 
void ComputeForwardDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, Context< T > *context, MatrixX< T > *J)
 
void ComputeCentralDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, Context< T > *context, MatrixX< T > *J)
 
void ComputeAutoDiffJacobian (const System< T > &system, const T &t, const VectorX< T > &xt, const Context< T > &context, MatrixX< T > *J)
 
void increment_num_iter_factorizations ()
 
void increment_jacobian_computation_derivative_evaluations (int count)
 
void increment_jacobian_evaluations ()
 
void set_jacobian_is_fresh (bool flag)
 
template<>
void ComputeAutoDiffJacobian (const System< AutoDiffXd > &, const AutoDiffXd &, const VectorX< AutoDiffXd > &, const Context< AutoDiffXd > &, MatrixX< AutoDiffXd > *)
 
- Protected Member Functions inherited from IntegratorBase< T >
const ContinuousState< T > & EvalTimeDerivatives (const Context< T > &context)
 Evaluates the derivative function and updates call statistics. More...
 
template<typename U >
const ContinuousState< U > & EvalTimeDerivatives (const System< U > &system, const Context< U > &context)
 Evaluates the derivative function (and updates call statistics). More...
 
void set_accuracy_in_use (double accuracy)
 Sets the working ("in use") accuracy for this integrator. More...
 
bool StepOnceErrorControlledAtMost (const T &h_max)
 Default code for advancing the continuous state of the system by a single step of h_max (or smaller, depending on error control). More...
 
CalcStateChangeNorm (const ContinuousState< T > &dx_state) const
 Computes the infinity norm of a change in continuous state. More...
 
std::pair< bool, T > CalcAdjustedStepSize (const T &err, const T &attempted_step_size, bool *at_minimum_step_size) const
 Calculates adjusted integrator step sizes toward keeping state variables within error bounds on the next integration step. More...
 
trajectories::PiecewisePolynomial< T > * get_mutable_dense_output ()
 Returns a mutable pointer to the internally-maintained PiecewisePolynomial instance, holding a representation of the continuous state trajectory since the last time StartDenseIntegration() was called. More...
 
bool DoDenseStep (const T &h)
 Calls DoStep(h) while recording the resulting step in the dense output. More...
 
ContinuousState< T > * get_mutable_error_estimate ()
 Gets an error estimate of the state variables recorded by the last call to StepOnceFixedSize(). More...
 
void set_actual_initial_step_size_taken (const T &h)
 
void set_smallest_adapted_step_size_taken (const T &h)
 Sets the size of the smallest-step-taken statistic as the result of a controlled integration step adjustment. More...
 
void set_largest_step_size_taken (const T &h)
 
void set_ideal_next_step_size (const T &h)
 

Constructor & Destructor Documentation

◆ ImplicitEulerIntegrator() [1/3]

◆ ImplicitEulerIntegrator() [2/3]

◆ ~ImplicitEulerIntegrator()

~ImplicitEulerIntegrator ( )
overridedefault

◆ ImplicitEulerIntegrator() [3/3]

ImplicitEulerIntegrator ( const System< T > &  system,
Context< T > *  context = nullptr 
)
explicit

Member Function Documentation

◆ get_error_estimate_order()

int get_error_estimate_order ( ) const
finalvirtual

Returns the asymptotic order of the difference between the large and small steps (from which the error estimate is computed), which is 2.

That is, the error estimate, ε* = x̅ⁿ⁺¹ − x̃ⁿ⁺¹ has the property that ‖ε*‖ = O(h²), and it deviates from the true error, ε, by ‖ε − ε*‖ = O(h³).

Derivation of the asymptotic order

This derivation is based on the same derivation for VelocityImplicitEulerIntegrator, and so the equation numbers are from there.

To derive the second-order error estimate, let us first define the vector-valued function e(tⁿ, h, xⁿ) = x̅ⁿ⁺¹ − xⁿ⁺¹, the local truncation error for a single, full-sized implicit Euler integration step, with initial conditions (tⁿ, xⁿ), and a step size of h. Furthermore, use to denote df/dt, and ∇f and ∇ẍ to denote the Jacobians df/dx and dẍ/dx of the ODE system ẋ = f(t, x). Note that uses a total time derivative, i.e., ẍ = ∂f/∂t + ∇f f.

Let us use x* to denote the true solution after a half-step, x(tⁿ+½h), and x̃* to denote the implicit Euler solution after a single half-sized step. Furthermore, let us use xⁿ*¹ to denote the true solution of the system at time t = tⁿ+h if the system were at x̃* when t = tⁿ+½h. See the following diagram for an illustration.

 Legend:
 ───── propagation along the true system
 :···· propagation using implicit Euler with a half step
 :---- propagation using implicit Euler with a full step

 Time  tⁿ         tⁿ+½h         tⁿ+h

 State :----------------------- x̅ⁿ⁺¹  <─── used for error estimation
       :
       :
       :
       :            :·········· x̃ⁿ⁺¹  <─── propagated result
       :            :
       :·········  x̃*   ─────── xⁿ*¹
       :
       xⁿ ───────  x*   ─────── xⁿ⁺¹  <─── true solution

We will use superscripts to denote evaluating an expression with x at that superscript and t at the corresponding time, e.g. ẍⁿ denotes ẍ(tⁿ, xⁿ), and f* denotes f(tⁿ+½h, x*). We first present a shortened derivation, followed by the longer, detailed version.

We know the local truncation error for the implicit Euler method is:

e(tⁿ, h, xⁿ) = x̅ⁿ⁺¹ − xⁿ⁺¹ = ½ h²ẍⁿ + O(h³).                  (10)

The local truncation error ε from taking two half steps is composed of these two terms:

e₁ = xⁿ*¹ − xⁿ⁺¹ = (1/8) h²ẍⁿ + O₁(h³),                       (15)
e₂ = x̃ⁿ⁺¹ − xⁿ*¹ = (1/8) h²ẍ* + O₂(h³) = (1/8) h²ẍⁿ + O₃(h³). (20)

In the long derivation, we will show that these second derivatives differ by at most O(h³).

Taking the sum,

ε = x̃ⁿ⁺¹ − xⁿ⁺¹ = e₁ + e₂ = (1/4) h²ẍⁿ + O(h³).               (21)

These two estimations allow us to obtain an estimation of the local error from the difference between the available quantities x̅ⁿ⁺¹ and x̃ⁿ⁺¹:

ε* = x̅ⁿ⁺¹ − x̃ⁿ⁺¹ = e(tⁿ, h, xⁿ) − ε,
                 = (1/4) h²ẍⁿ + O(h³),                        (22)

and therefore our error estimate is second order.

Below we will show this derivation in detail along with the proof that ‖ε − ε*‖ = O(h³):

Let us look at a single implicit Euler step. Upon Newton-Raphson convergence, the truncation error for implicit Euler is

e(tⁿ, h, xⁿ) = ½ h²ẍⁿ⁺¹ + O(h³)
             = ½ h²ẍⁿ + O(h³).                                (10)

To see why the two are equivalent, we can Taylor expand about (tⁿ, xⁿ),

ẍⁿ⁺¹ = ẍⁿ + h dẍ/dtⁿ + O(h²) = ẍⁿ + O(h).
e(tⁿ, h, xⁿ) = ½ h²ẍⁿ⁺¹ + O(h³) = ½ h²(ẍⁿ + O(h)) + O(h³)
             = ½ h²ẍⁿ + O(h³).

Moving on with our derivation, after one small half-sized implicit Euler step, the solution x̃* is

x̃* = x* + e(tⁿ, ½h, xⁿ)
   = x* + (1/8) h²ẍⁿ + O(h³),
x̃* − x* = (1/8) h²ẍⁿ + O(h³).                                 (11)

Taylor expanding about t = tⁿ+½h in this x = x̃* alternate reality,

xⁿ*¹ = x̃* + ½h f(tⁿ+½h, x̃*) + O(h²).                          (12)

Similarly, Taylor expansions about t = tⁿ+½h and the true solution x = x* also give us

xⁿ⁺¹ = x* + ½h f* + O(h²),                                    (13)
f(tⁿ+½h, x̃*) = f* + (∇f*) (x̃* − x*) + O(‖x̃* − x*‖²)
             = f* + O(h²),                                    (14)

where in the last line we substituted Eq. (11).

Eq. (12) minus Eq. (13) gives us,

xⁿ*¹ − xⁿ⁺¹ = x̃* − x* + ½h(f(tⁿ+½h, x̃*) − f*) + O(h³),
            = x̃* − x* + O(h³),

where we just substituted in Eq. (14). Finally, substituting in Eq. (11),

e₁ = xⁿ*¹ − xⁿ⁺¹ = (1/8) h²ẍⁿ + O(h³).                        (15)

After the second small step, the solution x̃ⁿ⁺¹ is

x̃ⁿ⁺¹ = xⁿ*¹ + e(tⁿ+½h, ½h, x̃*),
     = xⁿ*¹ + (1/8)h² ẍ(tⁿ+½h, x̃*) + O(h³).                   (16)

Taking Taylor expansions about (tⁿ, xⁿ),

x* = xⁿ + ½h fⁿ + O(h²) = xⁿ + O(h).                          (17)
x̃* − xⁿ = (x̃* − x*) + (x* − xⁿ) = O(h),                       (18)

where we substituted in Eqs. (11) and (17), and

ẍ(tⁿ+½h, x̃*) = ẍⁿ + ½h ∂ẍ/∂tⁿ + ∇ẍⁿ (x̃* − xⁿ) + O(h ‖x̃* − xⁿ‖)
             = ẍⁿ + O(h),                                     (19)

where we substituted in Eq. (18).

Substituting Eqs. (19) and (15) into Eq. (16),

x̃ⁿ⁺¹ = xⁿ*¹ + (1/8) h²ẍⁿ + O(h³)                              (20)
     = xⁿ⁺¹ + (1/4) h²ẍⁿ + O(h³),

therefore

ε = x̃ⁿ⁺¹ − xⁿ⁺¹ = (1/4) h² ẍⁿ + O(h³).                        (21)

Subtracting Eq. (21) from Eq. (10),

e(tⁿ, h, xⁿ) − ε = (½ − 1/4) h²ẍⁿ + O(h³);
⇒ ε* = x̅ⁿ⁺¹ − x̃ⁿ⁺¹ = (1/4) h²ẍⁿ + O(h³).                      (22)

Eq. (22) shows that our error estimate is second-order. Since the first term on the RHS matches ε (Eq. (21)),

ε* = ε + O(h³).                                               (23)

Implements IntegratorBase< T >.

◆ get_use_implicit_trapezoid_error_estimation()

bool get_use_implicit_trapezoid_error_estimation ( )

Returns true if the integrator will use implicit trapezoid for error estimation; otherwise it indicates the integrator will use step doubling for error estimation.

◆ operator=() [1/2]

ImplicitEulerIntegrator& operator= ( const ImplicitEulerIntegrator< T > &  )
delete

◆ operator=() [2/2]

ImplicitEulerIntegrator& operator= ( ImplicitEulerIntegrator< T > &&  )
delete

◆ set_use_implicit_trapezoid_error_estimation()

void set_use_implicit_trapezoid_error_estimation ( bool  flag)

Set this to true to use implicit trapezoid for error estimation; otherwise this integrator will use step doubling for error estimation.

By default this integrator will use step doubling.

◆ supports_error_estimation()

bool supports_error_estimation ( ) const
finalvirtual

Returns true, because this integrator supports error estimation.

Implements IntegratorBase< T >.


The documentation for this class was generated from the following file: