celeritas::DormandPrinceStepper< EquationT > Class Template Reference

Integrate the field ODEs using Dormand-Prince RK5(4)7M. More...

#include <DormandPrinceStepper.hh>

Public Types
Type aliases
using	result_type = FieldStepperResult

Public Member Functions
CELER_FUNCTION	DormandPrinceStepper (EquationT &&eq)
	Construct with the equation of motion.
CELER_FUNCTION result_type	operator() (real_type step, OdeState const &beg_state) const
	Numerically integrate using the DormandPrince RK5(4)7M method.

Detailed Description

template<class EquationT>
class celeritas::DormandPrinceStepper< EquationT >

Integrate the field ODEs using Dormand-Prince RK5(4)7M.

The algorithm, RK5(4)7M and the coefficients have been adapted from J. R. Dormand and P. J. Prince, "A family of embedded Runge-Kutta formulae" Journal Computational and Applied Mathematics, volume 6, no 1 (1980) and the coefficients to locate the mid point are taken from L. F. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics of * Computation, volume 46, number 17, pp 147 (1986).

For a given ordinary differential equation, \(dy/dx = f(x, y)\), the fifth order solution, \( y_{n+1} \), an embedded fourth order solution, \( y^{*}_{n+1} \), and the error estimate as difference between them are as follows,

\[ y_{n+1} = y_n + h \sum_{n=1}^{6} b_i k_i + O(h^6) y^{*}_{n+1} = y_n + h \sum_{n=1}^{7} b*_i k_i + O(h^5) y_{error} = y_{n+1} - y^{*}_{n+1} = \sum_{n=1}^{7} (b^{*}_i - b_i) k_i \]

where \(h\) is the step to advance and k_i is the right hand side of the function at \(x_n + h c_i\), and the coefficients (The Butcher table) for Dormand-Prince RK5(4)7M are

 c_i | a_ij
 0   |
 1/5 | 1/5
 3/10| 3/40        9/40
 4/5 | 44/45      -56/15      32/9
 8/9 | 19372/6561 -25360/2187 64448/6561 -212/729
 1   | 9017/3168  -355/33     46732/5247  49/176  -5103/18656
 1   | 35/384      0          500/1113    125/192 -2187/6784    11/84
----------------------------------------------------------------------------
 b*_i| 35/384      0          500/1113    125/192 -2187/6784    11/84    0
 b_i | 5179/57600  0          7571/16695  393/640 -92097/339200 187/2100 1/40

The result at the mid point is computed

\[ y_{n+1/2} = y_n + (h/2) \Sigma_{n=1}^{7} c^{*}_i k_i \]

with the coefficients \(c^{*}\) taken from L. F. Shampine (1986).

Todo:

Rename DormandPrinceIntegrator

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

• DormandPrinceStepper.hh