C1 DETEST

Equations:

$$\begin{align}\dot{x}_{0}&=- x_{0}\\\dot{x}_{1}&=x_{0} - x_{1}\\\dot{x}_{2}&=x_{1} - x_{2}\\\dot{x}_{3}&=x_{2} - x_{3}\\\dot{x}_{4}&=x_{3} - x_{4}\\\dot{x}_{5}&=x_{4} - x_{5}\\\dot{x}_{6}&=x_{5} - x_{6}\\\dot{x}_{7}&=x_{6} - x_{7}\\\dot{x}_{8}&=x_{7} - x_{8}\\\dot{x}_{9}&=x_{8}\end{align}$$

Description:

A radioactive decay chain with all constants equal to 1. Please note that the supplied exact solution is correct only for certain initial conditions (e.g., those from the IVPs).
Link to DETEST description

Keywords:

decoupled

Plots:

C6 (time to break-down)

C5 WPD

Initial value problems:

IVP 58

Class: PILB

Initial conditions:

$$\begin{align}x_{0}(0)&=1.0\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\\x_{3}(0)&=0.0\\x_{4}(0)&=0.0\\x_{5}(0)&=0.0\\x_{6}(0)&=0.0\\x_{7}(0)&=0.0\\x_{8}(0)&=0.0\\x_{9}(0)&=0.0\end{align}$$

Exact Solution:

$$\begin{align}x_{0}(t)&=1.0 \cdot e^{- t}\\x_{1}(t)&=1.0 \cdot t \cdot e^{- t}\\x_{2}(t)&=0.5 \cdot t^{2} \cdot e^{- t}\\x_{3}(t)&=0.166666666666667 \cdot t^{3} \cdot e^{- t}\\x_{4}(t)&=0.0416666666666667 \cdot t^{4} \cdot e^{- t}\\x_{5}(t)&=0.00833333333333333 \cdot t^{5} \cdot e^{- t}\\x_{6}(t)&=0.00138888888888889 \cdot t^{6} \cdot e^{- t}\\x_{7}(t)&=0.000198412698412698 \cdot t^{7} \cdot e^{- t}\\x_{8}(t)&=2.48015873015873 \cdot 10^{-5} \cdot t^{8} \cdot e^{- t}\\x_{9}(t)&=- 2.48015873015873 \cdot 10^{-5} \cdot \left(t^{8} + 8.0 \cdot t^{7} + 56.0 \cdot t^{6} + 336.0 \cdot t^{5} + 1680.0 \cdot t^{4} + 6720.0 \cdot t^{3} + 20160.0 \cdot t^{2} + 40320.0 \cdot t + 40320.0\right) \cdot e^{- t} + 1.0\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01960
c5: t_us=0.01770 , e_us=1.283786216506430e-15
c6: t_bd=10.000000 , e_bd=5.269118474870993e-13

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.02920
c5: t_us=0.02510 , e_us=8.865406814966090e-17
c6: t_bd=10.000000 , e_bd=2.498001805406602e-14

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.03620
c5: t_us=0.03140 , e_us=1.767334244446098e-18
c6: t_bd=10.000000 , e_bd=1.187938636348917e-14

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.57180
c5: t_us=0.56620 , e_us=7.184503812457822e-16
c6: t_bd=10.000000 , e_bd=6.971534460831208e-12

Generate graphics:

IVP 59

Class: PILB

Initial conditions:

$$\begin{align}x_{0}(0)&=\left[ 0.5, 1.5\right]\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\\x_{3}(0)&=0.0\\x_{4}(0)&=0.0\\x_{5}(0)&=0.0\\x_{6}(0)&=0.0\\x_{7}(0)&=0.0\\x_{8}(0)&=0.0\\x_{9}(0)&=0.0\end{align}$$

Exact Solution:

$$\begin{align}x_{0}(t)&=\left[0.5,1.5\right] \cdot e^{- t}\\x_{1}(t)&=\left[0.5,1.5\right] \cdot t \cdot e^{- t}\\x_{2}(t)&=0.5 \cdot \left[0.5,1.5\right] \cdot t^{2} \cdot e^{- t}\\x_{3}(t)&=0.166666666666667 \cdot \left[0.5,1.5\right] \cdot t^{3} \cdot e^{- t}\\x_{4}(t)&=0.0416666666666667 \cdot \left[0.5,1.5\right] \cdot t^{4} \cdot e^{- t}\\x_{5}(t)&=0.00833333333333333 \cdot \left[0.5,1.5\right] \cdot t^{5} \cdot e^{- t}\\x_{6}(t)&=0.00138888888888889 \cdot \left[0.5,1.5\right] \cdot t^{6} \cdot e^{- t}\\x_{7}(t)&=0.000198412698412698 \cdot \left[0.5,1.5\right] \cdot t^{7} \cdot e^{- t}\\x_{8}(t)&=2.48015873015873 \cdot 10^{-5} \cdot \left[0.5,1.5\right] \cdot t^{8} \cdot e^{- t}\\x_{9}(t)&=- 2.48015873015873 \cdot 10^{-5} \cdot \left[0.5,1.5\right] \cdot \left(t^{8} + 8.0 \cdot t^{7} + 56.0 \cdot t^{6} + 336.0 \cdot t^{5} + 1680.0 \cdot t^{4} + 6720.0 \cdot t^{3} + 20160.0 \cdot t^{2} + 40320.0 \cdot t + 40320.0\right) \cdot e^{- t} + \left[0.5,1.5\right]\end{align}$$

View test data:

Solver
Settings
Results

VNODE-LP

method: HOE
order: 10
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.01970
c5: t_us=0.01910 , e_us=1.125202607280207e-06
c6: t_bd=10.000000 , e_bd=6.671803212505651e-01

VNODE-LP

method: HOE
order: 15
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.02820
c5: t_us=0.02720 , e_us=1.125202598228484e-06
c6: t_bd=10.000000 , e_bd=6.671803212494318e-01

VNODE-LP

method: HOE
order: 20
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.03520
c5: t_us=0.03240 , e_us=1.125202597975911e-06
c6: t_bd=10.000000 , e_bd=6.671803212493616e-01

VNODE-LP

method: HOE
order: 5
abs. tolerance: 1e-12
rel. tolerance: 1e-12
min. stepsize: 1e-05

c4: t_c=0.62670
c5: t_us=0.62310 , e_us=1.125202643704065e-06
c6: t_bd=10.000000 , e_bd=6.671803212977596e-01

Generate graphics:

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