Comparison of:¶

(1) adaptive variable time steps
(2) fixed time steps
(3) exact solution

for the elementary reaction `A <-> B`,¶

taken to equilibrium.

This is a continuation of the experiments react_2_a (fixed time steps) and react_2_b (adaptive variable time steps)

Background: please see experiments react_2_a and react_2_b

TAGS : "numerical", "uniform compartment"¶

In [1]:

LAST_REVISED = "Jan. 3, 2026"
LIFE123_VERSION = "1.0.0rc7"        # Library version this experiment is based on

In [2]:

#import set_path              # Using MyBinder?  Uncomment this before running the next cell!

In [3]:

#import sys
#sys.path.append("C:/some_path/my_env_or_install")   # CHANGE to the folder containing your venv or libraries installation!
# NOTE: If any of the imports below can't find a module, uncomment the lines above, or try:  import set_path   

from life123 import check_version, ChemData, UniformCompartment

import numpy as np
import plotly.graph_objects as go
from life123 import ReactionKinetics, ReactionRegistry, PlotlyHelper

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Common set up for the chemicals and the reaction (used by all the simulations)¶

In [5]:

# Set up the reactions and their chemicals (common for all the simulations below)
rxns = ReactionRegistry()

rxns.add_reaction(reactants="A", products="B", kF=3., kR=2.)  # ELementary reaction A <-> B

rxns.describe_reactions()

add_reaction(): detected reaction type `ReactionUnimolecular`
Number of reactions: 1
0: A <-> B  Elementary Unimolecular reaction  (kF = 3 / kR = 2 / K = 1.5)
Chemicals involved in the above reactions: ['A', 'B']

In [ ]:

PART 1 - VARIABLE TIME STEPS¶

We'll do this part first, because the number of steps taken is unpredictable;
we'll note that number, and in Part 2 we'll do exactly that same number of fixed steps

In [6]:

dynamics_variable = UniformCompartment(reactions=rxns, preset="mid")

# Initial concentrations of all the chemicals
dynamics_variable.set_conc({"A": 10., "B": 50.})

dynamics_variable.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 10.0
  Species 1 (B). Conc: 50.0
Chemicals involved in reactions: ['B', 'A']

Run the reaction (VARIABLE adaptive time steps)¶

In [7]:

dynamics_variable.single_compartment_react(initial_step=0.1, target_end_time=1.2,
                                           variable_steps=True)
# Specifying variable_steps=True for emphasis; it's the default

19 total variable step(s) taken in 0.042 sec
Number of step re-do's because of elective soft aborts: 2
Norm usage: {'norm_A': 17, 'norm_B': 15, 'norm_C': 15, 'norm_D': 15}
System Time is now: 1.2268

The flag variable_steps automatically adjusts up or down the time steps¶

In part 2, we'll remember that it took 19 steps

In [8]:

dynamics_variable.get_history()   # The system's history, saved during the run of single_compartment_react()

Out[8]:

	SYSTEM TIME	A	B	step	caption
0	0.000000	10.000000	50.000000		Set concentration
1	0.016000	11.120000	48.880000	1	1st reaction step
2	0.032000	12.150400	47.849600	2
3	0.048000	13.098368	46.901632	3
4	0.067200	14.144925	45.855075	4
5	0.086400	15.091012	44.908988	5
6	0.109440	16.117327	43.882673	6
7	0.132480	17.025411	42.974589	7
8	0.160128	17.989578	42.010422	8
9	0.193306	18.986635	41.013365	9
10	0.233119	19.984624	40.015376	10
11	0.280894	20.943812	39.056188	11
12	0.338225	21.819882	38.180118	12
13	0.407022	22.569810	37.430190	13
14	0.489579	23.160168	36.839832	14
15	0.588647	23.576169	36.423831	15
16	0.707528	23.828097	36.171903	16
17	0.850186	23.950713	36.049287	17
18	1.021375	23.992900	36.007100	18
19	1.226802	24.000193	35.999807	19	last reaction step

In [9]:

dynamics_variable.plot_history(colors=['darkturquoise', 'green'], show_intervals=True)

Notice how the reaction proceeds in smaller steps in the early times, when [A] and [B] are changing much more rapidly¶

That resulted from passing the flag variable_steps=True to single_compartment_react()

In [ ]:

PART 2 - FIXED TIME STEPS¶

This is a re-do of the above simulation, but with a fixed time step¶

The fixed time step is chosen to attain the same total number of data points (i.e. 19) as obtained with the variable time steps of part 1

In [10]:

dynamics_fixed = UniformCompartment(reactions=rxns)   # Re-use the chemicals and reactions of part 1

In [11]:

# Initial concentrations of all the chemicals
dynamics_fixed.set_conc({"A": 10., "B": 50.})

dynamics_fixed.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 10.0
  Species 1 (B). Conc: 50.0
Chemicals involved in reactions: ['B', 'A']

Run the reaction (FIXED time steps)¶

In [12]:

# Matching the total number of steps to the earlier, variable-step simulation
dynamics_fixed.single_compartment_react(n_steps=19, target_end_time=1.2,
                                        variable_steps=False)

19 total fixed step(s) taken in 0.037 sec

In [13]:

dynamics_fixed.get_history()   # The system's history, saved during the run of single_compartment_react()

Out[13]:

	SYSTEM TIME	A	B	step	caption
0	0.000000	10.000000	50.000000		Set concentration
1	0.063158	14.421053	45.578947	1	1st reaction step
2	0.126316	17.445983	42.554017	2
3	0.189474	19.515673	40.484327	3
4	0.252632	20.931776	39.068224	4
5	0.315789	21.900689	38.099311	5
6	0.378947	22.563629	37.436371	6
7	0.442105	23.017220	36.982780	7
8	0.505263	23.327572	36.672428	8
9	0.568421	23.539917	36.460083	9
10	0.631579	23.685207	36.314793	10
11	0.694737	23.784615	36.215385	11
12	0.757895	23.852631	36.147369	12
13	0.821053	23.899169	36.100831	13
14	0.884211	23.931010	36.068990	14
15	0.947368	23.952796	36.047204	15
16	1.010526	23.967703	36.032297	16
17	1.073684	23.977902	36.022098	17
18	1.136842	23.984880	36.015120	18
19	1.200000	23.989655	36.010345	19	last reaction step

In [14]:

dynamics_fixed.plot_history(colors=['darkturquoise', 'green'], show_intervals=True)

Notice how grid points are being "wasted" on the tail part of the simulation, where little is happening - grid points that would be best used in the early part, as was done by the variable-step simulation of Part 1

In [ ]:

PART 3 - EXACT Solution (same fixed steps as in Part 2)¶

In [15]:

dynamics_exact = UniformCompartment(reactions=rxns, exact=True)   
# Re-use the chemicals and reactions of part 1 .  Note the `exact` flag

In [16]:

# Initial concentrations of all the chemicals
dynamics_exact.set_conc({"A": 10., "B": 50.})

dynamics_exact.describe_state()

SYSTEM STATE at Time t = 0:
2 species:
  Species 0 (A). Conc: 10.0
  Species 1 (B). Conc: 50.0
Chemicals involved in reactions: ['B', 'A']

Run the reaction (same FIXED time steps as before, but now using the exact solver)¶

In [17]:

# Matching the total number of steps to those of Part 1 and Part 2
dynamics_exact.single_compartment_react(n_steps=19, target_end_time=1.2,
                                        variable_steps=False)

19 total fixed step(s) taken in 0.036 sec

In [18]:

dynamics_exact.get_history()   # The system's history, saved during the run of single_compartment_react()

Out[18]:

	SYSTEM TIME	A	B	step	caption
0	0.000000	10.000000	50.000000		Set concentration
1	0.063158	13.791019	46.208981	1	1st reaction step
2	0.126316	16.555479	43.444521	2
3	0.189474	18.571359	41.428641	3
4	0.252632	20.041364	39.958636	4
5	0.315789	21.113312	38.886688	5
6	0.378947	21.894989	38.105011	6
7	0.442105	22.464999	37.535001	7
8	0.505263	22.880657	37.119343	8
9	0.568421	23.183761	36.816239	9
10	0.631579	23.404788	36.595212	10
11	0.694737	23.565964	36.434036	11
12	0.757895	23.683495	36.316505	12
13	0.821053	23.769200	36.230800	13
14	0.884211	23.831698	36.168302	14
15	0.947368	23.877272	36.122728	15
16	1.010526	23.910505	36.089495	16
17	1.073684	23.934739	36.065261	17
18	1.136842	23.952411	36.047589	18
19	1.200000	23.965297	36.034703	19	last reaction step

In [19]:

dynamics_exact.plot_history(colors=['darkturquoise', 'green'], show_intervals=True, title_prefix="EXACT solution")

In [ ]:

PART 4 - Comparing Variable Steps, Fixed Steps and Exact Solution¶

To avoid clutter, we'll just plot [A]¶

In [20]:

# A streamlined version of the diagram seen in Part 1
fig_variable = dynamics_variable.plot_history(chemicals='A', colors='darkturquoise', title="VARIABLE time steps", show=True)     

In [21]:

# A streamlined version of the diagram seen in Part 2
fig_fixed = dynamics_fixed.plot_history(chemicals='A', colors='blue', title="FIXED time steps", show=True)

In [22]:

# A streamlined version of the diagram seen in Part 3
fig_exact = dynamics_exact.plot_history(chemicals='A', colors='red', title="EXACT solution (at fixed time steps)", show=True)

In [23]:

PlotlyHelper.combine_plots(fig_list=[fig_fixed, fig_variable, fig_exact],
                           xrange=[0, 0.8], y_label="concentration [A]",
                           title="Variable time steps vs. Fixed vs. Exact soln, for [A] in reaction `A<->B`",
                           legend_title="Simulation run")    
# All the 3 plots put together: show only the initial part (but it's all there; you can zoom out!)

Not surprisingly, the adaptive variable time steps outperform the fixed ones (for the same total number of points in the time grid), in the early/middle part of the plot - i.e. at times when there's pronounced change.¶

All 3 curves essentially converge as the reaction approaches equilibrium.

In [ ]:

Comparison of:¶

for the elementary reaction A <-> B,¶

TAGS : "numerical", "uniform compartment"¶

Common set up for the chemicals and the reaction (used by all the simulations)¶

PART 1 - VARIABLE TIME STEPS¶

Run the reaction (VARIABLE adaptive time steps)¶

The flag variable_steps automatically adjusts up or down the time steps¶

Notice how the reaction proceeds in smaller steps in the early times, when [A] and [B] are changing much more rapidly¶

PART 2 - FIXED TIME STEPS¶

This is a re-do of the above simulation, but with a fixed time step¶

Run the reaction (FIXED time steps)¶

PART 3 - EXACT Solution (same fixed steps as in Part 2)¶

Run the reaction (same FIXED time steps as before, but now using the exact solver)¶

PART 4 - Comparing Variable Steps, Fixed Steps and Exact Solution¶

To avoid clutter, we'll just plot [A]¶

Not surprisingly, the adaptive variable time steps outperform the fixed ones (for the same total number of points in the time grid), in the early/middle part of the plot - i.e. at times when there's pronounced change.¶

for the elementary reaction `A <-> B`,¶