Download Quantifying the biological carbon pump

Integration schemes for biochemical systems
unconditional positivity and mass conservation
Jorn Bruggeman
Hans Burchard, Bob Kooi, Ben Sommeijer
Theoretical Biology
Vrije Universiteit, Amsterdam
Background
Master Theoretical biology (2003)
Start PhD study (2004)
“Understanding the ‘organic carbon pump’
in mesoscale ocean flows”
Focus: details in 1D water column
turbulence and biota, simulation in time
Tool: General Ocean Turbulence Model (GOTM)
modeling framework that hosts biota
Life is complex: aggregate!
individual
Kooijman (2000)
population
functional group
ecosystem


Aim: single model for population of ‘universal species’
One parameter per biological activity, e.g.
–
–

Bruggeman (2009)
nutrient affinity
detritus consumption
Parameter probability distributions = ecosystem biodiversity
Example

Functional group ‘phytoplankton’:
light
light harvesting
+
structural biomass
nutrient
+
nutrient uptake
maintenance

Start in end of winter:
–
–

deep mixed layer  little primary productivity
uniform trait distribution, low biomass for all ‘species’
No predation
Results
structural biomass
light harvesting biomass
nutrient harvesting biomass
Integration schemes

Biochemical criteria:
–
–
State variables remain positive
Elements and energy are conserved

Even if model meets criteria, integration results may not

GOTM: different schemes for different problems:
–
–
–
Advection (TVD schemes)
Diffusion (modified Crank-Nicholson scheme)
Production/destruction
Mass conservation

Model building block: reaction
r (...)
6 CO2  6 H 2 O 
 6 O2  1C6 H12O6

Conservation
–

Property of conservation
–
–

for any element, sums on left and right must be equal
is independent of r(…)
does depend on stoichiometric coefficients
Conservation = preservation of stoichiometric ratios
Systems of reactions


Integration scheme operates on ODEs
Reaction fluxes distributed over multiple ODEs:
dcCO2
r (...)
6 CO2  6 H 2O 
 6 O 2  C6 H12 O6
dt
dcH 2O
dt
dcO2
dt
dcC6 H12O6
dt
 6r (...)
 6r (...)

6r (...)

r (...)
Forward Euler, Runge-Kutta
cn 1  cn  t f  t n , c n 

Conservative
–


all fluxes multiplied with same factor Δt
Non-positive
Order: 1, 2, 4 etc.
Backward Euler, Gear
cn 1  cn  t f  t n 1 , c n 1 

Conservative
–



all fluxes multiplied with same factor Δt
Positive for order 1 (Hundsdorfer & Verwer)
Generalization to higher order eliminates positivity
Slow!
–
–
requires numerical approximation of partial derivatives
requires solving linear system of equations
Modified Patankar: concepts

Burchard, Deleersnijder, Meister (2003)
–

“A high-order conservative Patankar-type discretisation for stiff
systems of production-destruction equations”
Approach
–
–
–
–
Compound fluxes in production, destruction matrices (P, D)
Pij = rate of conversion from j to i
Dij = rate of conversion from i to j
Source fluxes in D, sink fluxes in P
Modified Patankar: structure
n 1
i
c



Flux-specific multiplication factors cn+1/cn
Represent ratio: (source after) : (source before)
Multiple sources in reaction:
–

 I  c nj 1  I
cin 1 
 c  t    Pij n     Dij  n 
 j 1 
c j  j 1
ci 
 
n
i
multiple, different cn+1/cn factors
Then: stoichiometric ratios not preserved!
Modified Patankar:
example/conclusion
r (...)
6 CO2  6 H 2O 
 6 O 2  C6 H12 O6
n 1


c
CO
n 1
n
2
cCO2  cCO2  t  6r (...) n 

cCO2 

n 1


c
H
n 1
n
2O
cH 2O  cH 2O  t  6r (...) n 

cH 2O 


2.


n
cCO
2

cHn 21O
cHn 2O
Conservative only if
1.

n 1
cCO
2
every reaction contains ≤ 1 source compound
source change ratios are identical (and remain so during simulation)
Positive
Order 1, 2
Requires solving linear system of equations
Typical MP conservation error
Total nitrogen over 20 years:
MP 1st order
600 % increase!
MP-RK 2nd order
New 1st order scheme: structure
cn1  cn  t f  t n , cn  p with
p
jJ n
c nj 1
c nj
J n  i : fi (t n , cn )  0, i {1,..., I }


Non-linear system of equations
Positivity requirement fixes domain of product term p:
p0
p 1


c nj


p  minn 
n
n
jJ 

 t  f j  t , c  
New 1st order scheme: solution

Non-linear system can be simplified to polynomial:
g ( p)   1  a j p   p  0 with a j 
t  f j  t n , c n 
jJ n

Polynomial in p:
–

positive at left bound p=0, negative at right bound
Derivative dg/dp < 0 within p domain:
–

c nj
only one valid p
Bisection technique is guaranteed to find p
Test case: linear system
Test case: non-linear system
New schemes: conclusion

Conservative
–
all fluxes multiplied with same factor pΔt

Positive
Extension to order 2 available

Relatively cheap

–
–
–

±20 bisection iterations = evaluations of polynomial
Always cheaper than Backward Euler
Cost scales with number of state variables, favorably compared
to Modified Patankar
Not for stiff systems (unlike Modified Patankar)
–
unless stiffness and positivity problems coincide
Plans

Publish new schemes
–

Short term
–
–
–

Bruggeman, Burchard, Kooi, Sommeijer (submitted 2005)
Explore trait-based models (different traits)
Trait distributions  single adapting species
Modeling coagulation (marine snow)
Extension to 3D global circulation models
The end
Test cases
Linear system:
Non-linear system: