runge_kutta_45 Derived Type

private subroutine oi_append_to_buffer(this, x, y, err)

Appends the supplied solution point to the internal solution buffer.

Arguments

private subroutine rk45_attempt_step(this, sys, h, x, y, f, yn, fn, yerr, k)

Attempts an integration step for this integrator.

Arguments

private subroutine oi_clear_buffer(this)

Clears the contents of the buffer.

Arguments

private pure function oi_estimate_error(this, y, yest, yerr) result(rst)

Computes the norm of the scaled error estimate. A value less than one indicates a successful step. A value greater than one suggests that the results do not meet the requested tolerances.

Arguments

Return Value real(kind=real64)

The norm of the scaled error.

private pure subroutine oi_initial_step(this, sys, xo, xf, yo, fo, h)

Computes an estimate of an initial step size.

Arguments

private function oi_next_step(this, e, eold, h, x, err) result(rst)

Estimates the next step size based upon the current and previous error estimates.

Arguments

Return Value real(kind=real64)

The new step size estimate.

private pure function oi_get_abs_tol(this) result(rst)

Gets the absolute error tolerance.

Arguments

Return Value real(kind=real64)

The tolerance value.

private pure function oi_get_allow_overshoot(this) result(rst)

Gets a value determining if the solver is allowed to overshoot the final value in the integration range.

Arguments

Return Value logical

True if the solver can overshoot, and then interpolate to achieve the required final value; else, false thereby indicating the solver cannot overshoot.

private pure function rk45_get_is_fsal(this) result(rst)

Gets a logical parameter stating if this is a first-same-as-last (FSAL) integrator.

Arguments

Return Value logical

True for a FSAL integrator; else, false.

private pure function oi_get_max_step(this) result(rst)

Gets the magnitude of the maximum allowed step size.

Arguments

Return Value real(kind=real64)

The step size limit.

private pure function oi_get_min_step(this) result(rst)

Gets the magnitude of the minimum allowed step size.

Arguments

Return Value real(kind=real64)

The step size limit.

private pure function rk45_get_order(this) result(rst)

Gets the order of the integrator.

Arguments

Return Value integer(kind=int32)

The order.

private pure function oi_get_abs_tol(this) result(rst)

Gets the absolute error tolerance.

Arguments

Return Value real(kind=real64)

The tolerance value.

private pure function oi_get_solution(this) result(rst)

Returns the solution computed by the integrator stored as a matrix with the first column containing the values of the independent variable at which the solution was computed. The remaining columns contain the solutions for each of the integrated equations in the order in which they appear in the source routine. Notice, the solve routine must be called before this routine.

Arguments

Return Value real(kind=real64), allocatable, dimension(:,:)

The resulting solution matrix.

private pure function rk45_get_stage_count(this) result(rst)

Gets the stage count for this integrator.

Arguments

Return Value integer(kind=int32)

The stage count.

private pure function oi_get_step_limit(this) result(rst)

Gets the limit on the number of integration steps that may be taken before the solver terminates.

Arguments

Return Value integer(kind=int32)

The step limit.

private pure function oi_get_control_parameter(this) result(rst)

Gets the step size control parameter $\beta$ used for PI control of the step size. A value of 0 provides a default step size controller (non-PI); however, a nonzero value of $\beta$ provides PI control that improves stability, but comes with a potential for efficiency loss. A good estimate for a starting point for this parameter is $\beta = \frac{0.4}{k}$ where $k$ is the order of the integrator.

The PI controller for step size is defined as follows. $$h_{n+1} = f_{s} h_{n} e_{n}^{-\alpha} e_{n-1}^{\beta}$$

Arguments

Return Value real(kind=real64)

The control parameter.

private pure function oi_get_safety_factor(this) result(rst)

Gets the safety factor (step size multiplier) used to provide a measure of control to the step size estimate such that $h_{new} = f_{s} h$, where $f_{s}$ is this safety factor.

Arguments

Return Value real(kind=real64)

The safety factor.

private subroutine rk45_interp(this, x, xn, yn, fn, xn1, yn1, fn1, y)

Performs the interpolation.

Arguments

private subroutine rk45_set_up_interp(this, sys, dense, x, xn, y, yn, f, fn, k)

Sets up the interpolation process.

Arguments

private subroutine rk45_pre_step(this, prevs, sys, h, x, y, f, err)

Placeholder routine for any pre-step actions.

Arguments

private subroutine oi_set_abs_tol(this, x)

Sets the absolute error tolerance.

Arguments

private subroutine oi_set_allow_overshoot(this, x)

Sets a value determining if the solver is allowed to overshoot the final value in the integration range.

Arguments

private subroutine oi_set_max_step(this, x)

Sets the magnitude of the maximum allowed step size.

Arguments

private subroutine oi_set_min_step(this, x)

Sets the magnitude of the minimum allowed step size.

Arguments

private subroutine oi_set_abs_tol(this, x)

Sets the absolute error tolerance.

Arguments

private subroutine oi_set_step_limit(this, x)

Sets the limit on the number of integration steps that may be taken before the solver terminates.

Arguments

private subroutine oi_set_control_parameter(this, x)

Sets the step size control parameter $\beta$ used for PI control of the step size. A value of 0 provides a default step size controller (non-PI); however, a nonzero value of $\beta$ provides PI control that improves stability, but comes with a potential for efficiency loss. A good estimate for a starting point for this parameter is $\beta = \frac{0.4}{k}$ where $k$ is the order of the integrator.

The PI controller for step size is defined as follows. $$h_{n+1} = f_{s} h_{n} e_{n}^{-\alpha} e_{n-1}^{\beta}$$

Arguments

private subroutine oi_set_safety_factor(this, x)

Sets the safety factor (step size multiplier) used to provide a measure of control to the step size estimate such that $h_{new} = f_{s} h$, where $f_{s}$ is this safety factor.

Arguments

private subroutine ssi_ode_solver(this, sys, x, iv, err)

Solves the supplied system of ODE's.

Arguments