Convergence analysis of implicit Runge-Kutta methods for ODE IVP#

Evaluation of relative quadrature error with the following parameters:

eq_names = [
    "y'' = -y"          ,
    "y'' = - exp(y)"    ,
    "y'' = xy"          ,
    "y' = Az; z' = By"  ,
]

method_names = [
    'Gauss'   ,
    'Radau_IA'   ,
    'Radau_IIA'   ,
    'Radau_IB'   ,
    'Radau_IIB'   ,
    'Lobatto_IIIA'   ,
    'Lobatto_IIIB'   ,
    'Lobatto_IIIC'   ,
    'Lobatto_IIIC*'  ,
    'Lobatto_IIID'  ,
    'Lobatto_IIIS'  ,
]

# all_orders = range(2,6)
all_orders = [4]

The following plots give the measured relative error as a function of the number of quadrature subintervals

plt.show()

Relative error on integrand y'' = -y, Relative error on integrand y'' = - exp(y), Relative error on integrand y'' = xy, Relative error on integrand y' = Az; z' = By

Error as a function of running time

plt.show()

Relative error as a function of computational cost for equation y'' = -y, Relative error as a function of computational cost for equation y'' = - exp(y), Relative error as a function of computational cost for equation y'' = xy, Relative error as a function of computational cost for equation y' = Az; z' = By

Total running time of the script: (0 minutes 2.499 seconds)

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