TimeStepping Namespace Reference

Classes
class	EmbeddedExplicitRungeKutta
class	ExplicitRungeKutta
class	ImplicitRungeKutta
class	LowStorageRungeKutta
class	RungeKutta
class	TimeStepping

Enumerations
enum	runge_kutta_method {   FORWARD_EULER , RK_THIRD_ORDER , SSP_THIRD_ORDER , RK_CLASSIC_FOURTH_ORDER ,   RK_FIFTH_ORDER , RK_SIXTH_ORDER , LOW_STORAGE_RK_STAGE3_ORDER3 , LOW_STORAGE_RK_STAGE5_ORDER4 ,   LOW_STORAGE_RK_STAGE7_ORDER4 , LOW_STORAGE_RK_STAGE9_ORDER5 , BACKWARD_EULER , IMPLICIT_MIDPOINT ,   CRANK_NICOLSON , SDIRK_TWO_STAGES , HEUN_EULER , BOGACKI_SHAMPINE ,   DOPRI , FEHLBERG , CASH_KARP , TSITOURAS5 ,   invalid }
enum	embedded_runge_kutta_time_step { DELTA_T , MIN_DELTA_T , MAX_DELTA_T }

Detailed Description

Namespace containing support for time stepping methods. The functionality of this namespace is used in step-67 and step-76, for example. For more advanced time stepping methods, see also the SUNDIALS namespace and the PETScWrappers::TimeStepper class as well as the step-86 tutorial program.

Enumeration Type Documentation

runge_kutta_method

enum TimeStepping::runge_kutta_method

The following Runge-Kutta methods are available:

• Explicit methods (see ExplicitRungeKutta::initialize):
  • FORWARD_EULER (first order)
  • HEUN_EULER (second order)
  • RK_THIRD_ORDER (third order Runge-Kutta)
  • SSP_THIRD_ORDER (third order SSP Runge-Kutta)
  • RK_CLASSIC_FOURTH_ORDER (classical fourth order Runge-Kutta)
  • RK_FIFTH_ORDER (six stages and fifth order)
  • RK_SIXTH_ORDER (seven stages and sixth order)
• Low-storage (explicit) Runge-Kutta methods (see LowStorageRungeKutta::initialize):
  • FORWARD_EULER (first order)
  • HEUN_EULER (second order)
  • LOW_STORAGE_RK_STAGE3_ORDER3 (three stages and third order)
  • LOW_STORAGE_RK_STAGE5_ORDER4 (five stages and fourth order)
  • LOW_STORAGE_RK_STAGE7_ORDER4 (seven stages and fourth order)
  • LOW_STORAGE_RK_STAGE9_ORDER5 (nine stages and fifth order)
• Implicit methods (see ImplicitRungeKutta::initialize):
  • BACKWARD_EULER (first order)
  • IMPLICIT_MIDPOINT (second order)
  • CRANK_NICOLSON (second order)
  • SDIRK_TWO_STAGES (second order)
• Embedded explicit methods (see EmbeddedExplicitRungeKutta::initialize):
  • HEUN_EULER (second order)
  • BOGACKI_SHAMPINE (third order)
  • DOPRI (Dormand-Prince method, fifth order; this is the method used by ode45 in MATLAB)
  • FEHLBERG (fifth order)
  • CASH_KARP (fifth order)
  • TSITOURAS5 (fifth order)

Enumerator
FORWARD_EULER	Forward Euler method, first order.
RK_THIRD_ORDER	Third order Runge-Kutta method.
SSP_THIRD_ORDER	Third order Strong Stability Preserving (SSP) Runge-Kutta method (SSP time discretizations are also called Total Variation Diminishing (TVD) methods in the literature, see [111]).
RK_CLASSIC_FOURTH_ORDER	Classical fourth order Runge-Kutta method.
RK_FIFTH_ORDER	Fifth order Runge-Kutta method.
RK_SIXTH_ORDER	Sixth order Runge-Kutta method.
LOW_STORAGE_RK_STAGE3_ORDER3	Three-stage scheme of order three by Kennedy et al. [146]. Its stability region is significantly smaller than the higher order schemes, but due to three stages only, it is very competitive in terms of the work per stage.
LOW_STORAGE_RK_STAGE5_ORDER4	Five-stage scheme of order four, defined in the paper by Kennedy et al. [146].
LOW_STORAGE_RK_STAGE7_ORDER4	Seven-stage scheme of order four defined in the paper by Tselios and Simos [228].
LOW_STORAGE_RK_STAGE9_ORDER5	Nine-stage scheme of order five defined in the paper by Kennedy et al. [146].
BACKWARD_EULER	Backward Euler method, first order.
IMPLICIT_MIDPOINT	Implicit midpoint method, second order.
CRANK_NICOLSON	Crank-Nicolson method, second order.
SDIRK_TWO_STAGES	Two stage SDIRK method (short for "singly diagonally implicit Runge-Kutta"), second order.
HEUN_EULER	Heun's method (improved Euler's method), second order.
BOGACKI_SHAMPINE	Bogacki–Shampine method, third-order.
DOPRI	Dormand-Prince method, fifth order; this is the method used by ode45 in MATLAB.
FEHLBERG	Fehlberg method, fifth order.
CASH_KARP	Cash–Karp method, fifth order.
TSITOURAS5	Tsitouras method, fifth order.
invalid	Invalid.

Definition at line 67 of file time_stepping.h.

embedded_runge_kutta_time_step

enum TimeStepping::embedded_runge_kutta_time_step

Reason for exiting evolve_one_time_step when using an embedded method: DELTA_T, MIN_DELTA_T, MAX_DELTA_T.

Enumerator
DELTA_T	The time step is in the valid range.
MIN_DELTA_T	The time step was increased to the minimum acceptable time step.
MAX_DELTA_T	The time step was reduced to the maximum acceptable time step.

Definition at line 171 of file time_stepping.h.