The increasing impacts of climate change require strategies for climate adaptation. Dynamic global vegetation models (DGVMs) are one type of multi-sectorial impact model with which the effects of multiple interacting processes in the terrestrial biosphere under climate change can be studied. The complexity of DGVMs is increasing as more and more processes, especially for plant physiology, are implemented. Therefore, there is a growing demand for increasing the computational performance of the underlying algorithms as well as ensuring their numerical accuracy. One way to approach this issue is to analyse the routines which have the potential for improved computational efficiency and/or increased accuracy when applying sophisticated mathematical methods.

In this paper, the Farquhar–Collatz photosynthesis model under water stress as implemented in the Lund–Potsdam–Jena managed Land DGVM (4.0.002) was examined.
We additionally tested the uncertainty of most important parameter of photosynthesis as an additional approach to improve model quality. We found that the numerical solution of a nonlinear equation, so far solved with the bisection method, could be significantly improved by using Newton's method instead. The latter requires the computation of the derivative of the underlying function which is presented. Model simulations show a significantly lower number of iterations to solve the equation numerically and an overall run time
reduction of the model of about 16 % depending on the chosen accuracy.
Increasing the parameters

Climate change is increasingly affecting the world we live in, and that in turn affects nature's contribution to our livelihoods

The Farquhar–Collatz approach was implemented in the land surface of the SiB2 model by

While land surface models detail vertical water, energy, and carbon profiles within the canopy, which extrapolates the photosynthetic capacity calculated at the leaf level to canopy photosynthesis

In order to apply the model to the global land surface it is no longer sufficient to use faster or larger computing infrastructure or to try to parallelise the code as in

To illustrate our approach, our goal was to improve the computational efficiency of DGVMs by accelerating the photosynthesis module under water stress conditions using the Lund–Potsdam–Jena DGVM,

We start with a short description of the different mathematical methods to find the zeros of a general nonlinear continuous function

The computation of the ratio

Here, the computational efficiency is determined by the speed of convergence. To compare the methods with respect to the speed of convergence we define the order of convergence as follows:
let

Let us introduce some of the methods in the following subsections, see

For bisection we have to choose

For the regula falsi method, we also need to choose

The secant method only differs from the regula falsi in that the starting values

Newton's method starts at an arbitrary approximation

We now analyse the difference in speed of convergence between the bisection and Newton's methods when applied to the optimisation equation of the photosynthesis routine of the LPJmL DGVM.

In presenting the function

To shorten the formulas we define the abbreviation

Light-limited photosynthesis depends on the absorbed photosynthetically active radiation (APAR); Rubisco-limited photosynthesis is determined by the maximum Rubisco capacity

To compute the derivative

The function

The condition

Function

We have tested the different methods in the routine regarding computational time and number of iterations for given accuracy

In a first test, the LPJmL model was run over 120 simulation years and the number of iterations in the bisection and Newton's routine was counted and averaged over all grid cells and one year (Fig.

In a next step, a spin-up run of LPJmL over 5000 simulation years was conducted to compare the time performance using both routines. Usually, LPJmL simulation experiments start from bare ground, i.e. initial vegetation conditions are not prescribed. Therefore, a spin-up run is used to bring all vegetation and soil carbon pools into equilibrium with climate.
For the usually implemented accuracy

Average number of iteration for bisection (upper lines, blue) and Newton (lower lines, red) for accuracy

Mean decadic logarithm of the accuracy

In order to check if the implementation of Newton's method is robust for all important model variables, we performed a transient simulation with the LPJmL model starting from the spin-up and covering the years 1901–2000. Model configuration and input data are as in

Parameter sensitivity on annual gross primary productivity (AGPP, average of 1901–2000) shown as the difference between new parameter and reference simulations. Both simulations have the Newton approach implemented. Increasing

The photosynthesis module is also applied to the crop functional types and managed grassland within LPJmL4.0. Therefore, sawing dates, crop productivity, and harvest are among the simulated variables. Comparing both model versions in the model benchmark, we found that global harvest changed for a number of crops. Rainfed and irrigated rice increased by 5 % and 8 %, respectively, mainly in India and southeast Asia. Harvest of rainfed temperate cereals increased by

For all global carbon pools (vegetation and soil) and carbon (GPP, heterotrophic respiration, and fire emissions) as well as water fluxes (transpiration and runoff) we found no difference in the temporal changes in the transient simulation over the 20th century. All variables showed similar, if not identical, dynamics (data not shown).
Small changes were found in the fractional coverage of plant functional types, i.e. most differences were negligible. The fractional coverage of temperate broadleaved summergreen trees increased by 4.8 % globally, which mainly applies to Europe, the northeastern USA, and parts of China. Increases in temperate C

After improving the computational efficiency and numerical precision, we can now test the parameter uncertainties following

Change in the AGPP after varying the listed parameters by 10 %. GPP is calculated as the global average mean for the years 1901–2000.

Geographically, increasing

We remark that future work on the photosynthesis approach could focus on the new Johnson and Berry scheme

The computational load of dynamic global vegetation models, caused by increased complexity of the modelling processes, has so far been counteracted by the high-performance computing systems used. However, more recently it has become clear that updates in computing infrastructure are not sufficient anymore. Consequently, we proposed to carefully evaluate the algorithmic structure of DGVMs and identify and update routines that can benefit from the use of modern mathematical methods. As a showcase, we investigated the photosynthesis model in the LPJmL DGVM. Specifically, we investigated the computation of the ratio

General parameters used in the photosynthesis routine. PFT is plant functional type.

PFT-specific parameter for temperature stress function (Eq. 12) in

To implement Newton's method in the LPJmL code, changes had to be made in the functions

New function

The benchmark table of global status variables (Table

Global sums of actual vegetation, including land-use, comparing Newton approach (benchmark run) against bisection approach (run). Tece is temperate cereals. NA – not applicable, Mha – megahectare, Mt DM – megatonnes of dry matter.

Continued.

Literature:

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The model code is available at

The supplement related to this article is available online at:

JN and RR performed the mathematical analysis, JN and WvB implemented and tested the new numerical methods, and WvB conducted the simulation experiments and analysed the model performance and computation efficiency. JN and KT wrote the paper and all authors contributed to the writing of the paper and discussion of the model study throughout to develop the work.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research, and the Land Brandenburg for supporting this project by providing resources on the high-performance computer system at the Potsdam Institute for Climate Impact Research. We thank Marie Hemmen from PIK for her support in benchmarking the LPJmL model.

The high-performance computing system at PIK was funded by the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research, and the Land Brandenburg.

This paper was edited by Carlos Sierra and reviewed by two anonymous referees.