THE DANGER OF EXCESSIVELY LARGE SINGLE TIME STEPS IN DIFFUSION¶

-> When the time step = (0.5 / diffusion rate),
a 2-bin system equilibrates in a single step,
and some 3-bin systems can over-shoot equilibrium!

-> When the time step = (0.33333 / diffusion rate),
some 3-bin systems equilibrate in a single step

So, (0.33333 / diffusion rate) is a - rather lax - upper bound for sensible single time steps!

IMPORTANT: The above is for delta_x = 1; in general, multiply by delta_x**2

The "Von Neumann stability analysis", which provides a slighly looser max value of time steps, is also discussed.

That value of 0.33333 is saved in the Class variable "time_step_threshold"

TAGS : "diffusion 1D", "under-the-hood"¶

In [1]:

LAST_REVISED = "May 27, 2025"
LIFE123_VERSION = "1.0.0rc3"        # 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, os
#os.getcwd()
#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 ChemData, BioSim1D, check_version

In [4]:

check_version(LIFE123_VERSION)

OK

In [ ]:

Simulate a 2-bin system, with 1 chemical¶

In [5]:

chem_data = ChemData(diffusion_rates=10.)

In [6]:

bio = BioSim1D(n_bins=2, chem_data=chem_data)
bio.inject_conc_to_bin(bin_address=0, delta_conc=100., chem_index=0)
bio.describe_state()

SYSTEM STATE at Time t = 0:
2 bins and 1 chemical species

Out[6]:

	Species	Diff rate	Bin 0	Bin 1
0	A	10.0	100.0	0.0

In [7]:

bio.time_step_threshold = 0.51    # TO BYPASS THE SAFETY LIMIT 
                                  # (for the allowed maximum delta Time) that is typically in place

# When the time step is (0.5 / diffusion rate),
# a 2-bin system equilibrates in a single step!
bio.diffuse(time_step=0.05, n_steps=1)
bio.describe_state()

SYSTEM STATE at Time t = 0.05:
2 bins and 1 chemical species

Out[7]:

	Species	Diff rate	Bin 0	Bin 1
0	A	10.0	50.0	50.0

Note the [50. 50.] values : the two bins have equilibratedin a single simulation step!
Mighty suspicious!!

In [ ]:

Start over with a 3-bin system¶

In [8]:

bio = BioSim1D(n_bins=3, chem_data=chem_data)
bio.inject_conc_to_bin(bin_address=1, delta_conc=100., chem_index=0)
bio.describe_state()

SYSTEM STATE at Time t = 0:
3 bins and 1 chemical species

Out[8]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	0.0	100.0	0.0

In [9]:

bio.time_step_threshold = 0.51    # TO BYPASS THE SAFETY LIMIT 
                                  # (for the allowed maximum delta Time) that is typically in place

# When the time step is (0.5 / diffusion rate),
# a 3-bin system can overshoot equilibrium!
bio.diffuse(time_step=0.05, n_steps=1)
bio.describe_state()

SYSTEM STATE at Time t = 0.05:
3 bins and 1 chemical species

Out[9]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	50.0	0.0	50.0

Note the concentrations of [50. 0. 50.] : the diffusion has over-shot equilibrium!!!

Cleary, a very excessively large time step!¶

In [ ]:

Start over again with a new 3-bin system¶

In [10]:

bio = BioSim1D(n_bins=3, chem_data=chem_data)
bio.inject_conc_to_bin(bin_address=1, delta_conc=100., chem_index=0)
bio.describe_state()

SYSTEM STATE at Time t = 0:
3 bins and 1 chemical species

Out[10]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	0.0	100.0	0.0

In [11]:

bio.time_step_threshold = 0.34    # TO slightly BYPASS THE SAFETY LIMIT 
                                  # (for the allowed maximum delta Time) that is typically in place

bio.diffuse(time_step=0.033333, n_steps=1)
bio.describe_state()

SYSTEM STATE at Time t = 0.033333:
3 bins and 1 chemical species

Out[11]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	33.333	33.334	33.333

When the time step is (0.33333 / diffusion rate),¶

a 3-bin system, configured as above, equilibrates in a single step!¶

Again, an excessively large time step! (note that we're using the default value of 1 for delta_x)¶

Start over again with a new 3-bin system : same as the last round, but this time use a delta_x = 10 (rather than the default 1)¶

In [12]:

bio = BioSim1D(n_bins=3, chem_data=chem_data)
bio.inject_conc_to_bin(bin_address=1, delta_conc=100., chem_index=0)
bio.describe_state()

SYSTEM STATE at Time t = 0:
3 bins and 1 chemical species

Out[12]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	0.0	100.0	0.0

In [13]:

bio.time_step_threshold = 0.34    # TO slightly BYPASS THE SAFETY LIMIT 
                                  # (for the allowed maximum delta Time) that is typically in place
    
# When the time step is (delta_x**2 * 0.33333 / diffusion rate),
# a 3-bin system, configured as above, equilibrates in a single step!

In [14]:

bio.diffuse(time_step=3.3333, n_steps=1, delta_x=10)
bio.describe_state()

SYSTEM STATE at Time t = 3.3333:
3 bins and 1 chemical species

Out[14]:

	Species	Diff rate	Bin 0	Bin 1	Bin 2
0	A	10.0	33.333	33.334	33.333

This generalizes the previous result to any arbitrary delta_x :¶

in this scenario, a time step of (delta_x*2 0.33333 / diffusion rate)¶

lead to equilibrium in a single step!¶

In [ ]:

The physical intuition explored in this experiment suggests enforcing:¶

`delta_t < delta_x**2 * 0.33333 / diffusion rate`¶

Interestingly, this is only slightly stricter than the upper bound that emerges from the "Von Neumann stability analysis" of the diffusion equation in 1-D, which states that solutions may become unstable unless¶

delta_t < delta_x*2 0.5 / diffusion rate

NOTE: The method `BioSim1D.diffuse_step()` enforces the stricter upper bound¶

To keep in mind that the "Von Neumann stability analysis" is ONLY applicable when the diffusion equation is being solved with the "explicit Forward-Time Centered Space" method, which is the one currently used by the `diffuse_step()` method.¶

An explanation can be found at: https://www.youtube.com/watch?v=QUiUGNwNNmo

In [ ]:

THE DANGER OF EXCESSIVELY LARGE SINGLE TIME STEPS IN DIFFUSION¶

TAGS : "diffusion 1D", "under-the-hood"¶

Simulate a 2-bin system, with 1 chemical¶

Start over with a 3-bin system¶

Cleary, a very excessively large time step!¶

Start over again with a new 3-bin system¶

When the time step is (0.33333 / diffusion rate),¶

a 3-bin system, configured as above, equilibrates in a single step!¶

Again, an excessively large time step! (note that we're using the default value of 1 for delta_x)¶

Start over again with a new 3-bin system : same as the last round, but this time use a delta_x = 10 (rather than the default 1)¶

This generalizes the previous result to any arbitrary delta_x :¶

in this scenario, a time step of (delta_x*2 0.33333 / diffusion rate)¶

lead to equilibrium in a single step!¶

The physical intuition explored in this experiment suggests enforcing:¶

delta_t < delta_x**2 * 0.33333 / diffusion rate¶

Interestingly, this is only slightly stricter than the upper bound that emerges from the "Von Neumann stability analysis" of the diffusion equation in 1-D, which states that solutions may become unstable unless¶

NOTE: The method BioSim1D.diffuse_step() enforces the stricter upper bound¶

To keep in mind that the "Von Neumann stability analysis" is ONLY applicable when the diffusion equation is being solved with the "explicit Forward-Time Centered Space" method, which is the one currently used by the diffuse_step() method.¶

`delta_t < delta_x**2 * 0.33333 / diffusion rate`¶

NOTE: The method `BioSim1D.diffuse_step()` enforces the stricter upper bound¶

To keep in mind that the "Von Neumann stability analysis" is ONLY applicable when the diffusion equation is being solved with the "explicit Forward-Time Centered Space" method, which is the one currently used by the `diffuse_step()` method.¶