Introduction

Implicit Runge–Kutta methods have a number of desirable properties.

The Gauss–Legendre methods, for example, are self-adjoint, meaning that they provide the same solution when integrating forward or backward in time.

Coefficients

A generic framework for implicit Runge–Kutta methods has been implemented. The focus so far is on methods with interesting geometric properties and currently covers the following schemes:

• "ImplicitRungeKuttaGaussCoefficients"

• "ImplicitRungeKuttaLobattoIIIACoefficients"

• "ImplicitRungeKuttaLobattoIIIBCoefficients"

• "ImplicitRungeKuttaLobattoIIICCoefficients"

• "ImplicitRungeKuttaRadauIACoefficients"

• "ImplicitRungeKuttaRadauIIACoefficients"

The derivation of the method coefficients can be carried out to arbitrary order and arbitrary precision.

Coefficient Generation

• Start with the definition of the polynomial, defining the abscissas of the stage coefficients. For example, the abscissas for Gauss–Legendre methods are defined as .

• Univariate polynomial factorization gives the underlying irreducible polynomials defining the roots of the polynomials.

• Root objects are constructed to represent the solutions (using unique root isolation and Jenkins–Traub for the numerical approximation).

• Root objects are then approximated numerically for precision coefficients.

• Condition estimates for Vandermonde systems governing the coefficients yield the precision to take in approximating the roots numerically.

• Specialized solvers for nonconfluent Vandermonde systems are then used to solve equations for the coefficients (see [GVL96]).

• One step of iterative refinement is used to polish the approximate solutions and to check that the coefficients are obtained to the requested precision.
This generates the coefficients for the two-stage fourth-order Gauss–Legendre method to 50 decimal digits of precision:

The coefficients have the form .

This generates the coefficients for the two-stage fourth-order Gauss–Legendre method exactly. For high-order methods, generating the coefficients exactly can often take a very long time:
This generates the coefficients for the six-stage tenth-order RaduaIA implicit Runge–Kutta method to 20 decimal digits of precision:

Examples

The "ImplicitSolver" method of "ImplicitRungeKutta" has options AccuracyGoal and PrecisionGoal that specify the absolute and relative error to aim for in solving the nonlinear system of equations.

These options have the same default values as the corresponding options in NDSolve, since often there is little point in solving the nonlinear system to much higher accuracy than the local error of the method.

The first invariant is the Hamiltonian of the system, and the error is now bounded, as it should be, since the Gauss implicit Runge–Kutta method is a symplectic integrator.

This defines the implicit midpoint method as the one-stage implicit Runge–Kutta method of order two.

At present, the implicit Runge–Kutta method framework does not use banded Newton techniques for uncoupling the nonlinear system.

Option Summary

"ImplicitRungeKutta" Options

option name	default value
"Coefficients"	"ImplicitRungeKuttaGaussCoefficients"	specify the coefficients to use in the implicit Runge–Kutta method
"DifferenceOrder"	Automatic	specify the order of local accuracy of the method
"ImplicitSolver"	"Newton"	specify the solver to use for the nonlinear system; valid settings are FixedPoint or "Newton"
"StepSizeControlParameters"	Automatic	specify the step control parameters
"StepSizeRatioBounds"	{,4}	specify the bounds on a relative change in the new step size
"StepSizeSafetyFactors"	Automatic	specify the safety factors to use in the step size estimate

Options of the method "ImplicitRungeKutta".

The default setting of Automatic for the option "StepSizeSafetyFactors" uses the values {9/10,9/10}.

"ImplicitSolver" Options

option name	default value
AccuracyGoal	Automatic	specify the absolute tolerance to use in solving the nonlinear system
"IterationSafetyFactor"		specify the safety factor to use in solving the nonlinear system
MaxIterations	Automatic	specify the maximum number of iterations to use in solving the nonlinear system
PrecisionGoal	Automatic	specify the relative tolerance to use in solving the nonlinear system

Common options of "ImplicitSolver".

option name	default value
"JacobianEvaluationParameter"		specify when to recompute the Jacobian matrix in Newton iterations
"LinearSolveMethod"	Automatic	specify the linear solver to use in Newton iterations
"LUDecompositionEvaluationParameter"		specify when to compute LU decompositions in Newton iterations

Options specific to the "Newton" method of "ImplicitSolver".