Application of modified Patankar schemes to stiff biogeochemical models for the water column

Abstract

In this paper, we apply recently developed positivity preserving and conservative Modified Patankar-type solvers for ordinary differential equations to a simple stiff biogeochemical model for the water column. The performance of this scheme is compared to schemes which are not unconditionally positivity preserving (the first-order Euler and the second- and fourth-order Runge–Kutta schemes) and to schemes which are not conservative (the first- and second-order Patankar schemes). The biogeochemical model chosen as a test ground is a standard nutrient–phytoplankton–zooplankton–detritus (NPZD) model, which has been made stiff by substantially decreasing the half saturation concentration for nutrients. For evaluating the stiffness of the biogeochemical model, so-called numerical time scales are defined which are obtained empirically by applying high-resolution numerical schemes. For all ODE solvers under investigation, the temporal error is analysed for a simple exponential decay law. The performance of all schemes is compared to a high-resolution high-order reference solution. As a result, the second-order modified Patankar–Runge–Kutta scheme gives a good agreement with the reference solution even for time steps 10 times longer than the shortest numerical time scale of the problem. Other schemes do either compute negative values for non-negative state variables (fully explicit schemes), violate conservation (the Patankar schemes) or show low accuracy (all first-order schemes).

Appendices

Appendix 1

Discrete schemes for reaction terms

In this section, the ODE solvers used in the present investigation are given. Here, $\underline{c}^{n}=(c_{i}^{n})_{i=1,\ldots,I}$ represents the discrete result vector at the old time step and $\underline{c}^{n+1}=(c_{i}^{n+1})_{i=1,\ldots,I}$ at the new time step.

Conservative schemes

First-order Euler-forward (E1)

$$ c_{i}^{n+1} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\underline{c}^{n}\right) - D_{i}\left(\underline{c}^n\right) \right\}.$$

(21)

Second-order Runge–Kutta (RK2)

$$\left. { \begin{array}{*{20}l} c^{(1)}_{i} = c^{n}_{i} + \Delta t \left\{ {P_{i} (\underline{c}^{n} ) - D_{i} (\underline{c}^{n} )} \right\}, \\ c^{n + 1}_{i} = c^{n}_{i} + \frac{\Delta t}{2} \left\{P_{i} (\underline{c}^{n}) + P_{i} \left( {\underline{c}^{(1)} } \right) - D_{i} (\underline{c} ^{n} ) - D_{i} \left( {\underline{c} ^{(1)} } \right)\right\}. \\ \end{array} } \right\}$$

(22)

Fourth-order Runge–Kutta (RK4)

$$ \left. {\begin{array}{*{20}l} c_{i}^{(1)} = c_{i}^{n} + \Delta t \left\{P_{i}(\underline{c}^{n}) - D_{i}(\underline{c}^{n})\right\} \\ c_{i}^{(2)} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\frac{1}{2}\left(\underline{c}^{n} + \underline{c}^{(1)}\right)\right) - D_{i}\left(\frac{1}{2}\left(\underline{c}^{n} + \underline{c}^{(1)}\right)\right)\right\} \\ c_{i}^{(3)} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\frac{1}{2}\left(\underline{c}^{n} + \underline{c}^{(2)} \right)\right) - D_{i}\left(\frac{1}{2}\left(\underline{c}^{n} + \underline{c}^{(2)}\right)\right)\right\}\\ c_{i}^{(4)} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\underline{c}^{(3)}\right) - D_{i}\left(\underline{c}^{(3)}\right)\right\}\\ c_{i}^{n+1} = \frac{1}{6} \left\{c_{i}^{(1)} + 2c_{i}^{(2)} + 2c_{i}^{(3)} + c_{i}^{(4)} \right\}.\\ \end{array} } \right\}$$

(23)

Positivity-preserving schemes

First-order Patankar–Euler scheme (PE1)

$$ c_{i}^{n+1} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\underline{c}^{n}\right) - D_{i}\left(\underline{c}^{n}\right) \frac{{c_{i}^{n+1}}}{{c_{i}^{n}}} \right\}.$$

(24)

Second-order Patankar–Runge–Kutta (PRK2)

$$ \left. {\begin{array}{*{20}l} c_{i}^{(1)} = c_{i}^{n} + \Delta t \left\{P_{i}\left(\underline{c}^{n}\right) - D_{i}\left(\underline {c}^n\right) \frac{{c_i^{(1)}}}{{c_i^n}}\right\},\\ c_{i}^{n+1} = c_{i}^{n} + \frac{{\Delta t}}{{2}} \left\{P_{i}\left(\underline{c}^n\right) + P_{i}\left(\underline{c}^{(1)}\right) - \left(D_{i}\left(\underline{c}^{n}\right) + D_{i}\left(\underline{c}^{(1)}\right)\right) \frac{{c_i^{n+1}}}{{c_i^{(1)}}} \right\}.\\ \end{array} } \right\}$$

(25)

Conservative and positivity-preserving schemes

First-order modified Patankar–Euler scheme (MPE1)

$$ c_{i}^{n+1} = c_{i}^{n} + \Delta t \left\{\sum\limits_{j=1}^{I} p_{i,j}\left(\underline{c}^{n}\right) \frac{{c_{j}^{n+1}}}{{c_{j}^{n}}} - \sum\limits_{j=1}^{I} d_{i,j}\left(\underline{c}^{n}\right)\frac{{c_{i}^{n+1}}}{{c_{i}^{n}}} \right\}.$$

(26)

Second-order modified Patankar–Runge–Kutta (MPRK2)

$$ \left. {\begin{array}{*{20}l} c_{i}^{(1)} = c_{i}^{n} + \Delta t \left\{\sum\limits_{j=1}^{I} p_{i,j}\left(\underline{c}^{n}\right)\frac{{c_{j}^{(1)}}}{ {c_{j}^{n}}} - \sum\limits_{j=1}^{I} d_{i,j}\left(\underline{c}^{n}\right) \frac{{c_i^{(1)}}}{{c_i^n}}\right\},\\ c_{i}^{n+1} = c_{i}^{n} + \frac{{\Delta t}}{{2}} \left\{\sum\limits_{j=1}^{I} \left(p_{i,j}\left(\underline{c}^{n}\right) + p_{i,j}\left(\underline{c}^{(1)}\right) \right) \frac{{c_{j}^{n+1}}}{{c_{j}^{(1)}}} - \sum\limits_{j=1}^{I} \left(d_{i,j}\left(\underline{c}^{n}\right) + d_{i,j}\left(\underline{c}^{(1)}\right) \right) \frac{{c_{i}^{n+1}}}{{c_{i}^{(1)}}} \right\}.\\ \end{array} } \right\}$$

(27)

Appendix 2

NPZD model details

The NPZD model described in Sect. NPZD model consists of I=4 state variables, see Table 3.

Table 3 State variables of the NPZD model

Seven processes expressed as sink terms are included in this conservative model, see Eqs. 28, 29, 30, 31, 32, 33, 34, 35.

Nutrient uptake by phytoplankton

$$d_{np}=r_{\max} \frac{{I_{\rm PAR}}}{I_{\rm opt}}\exp \left(1-\frac{{I_{\rm PAR}}}{{I_{\rm opt}}}\right)\frac{{c_n}}{{\alpha+c_{n}}}c_{p}$$

(28)

with

$$I_{\rm opt}=\max\left(\frac{1}{4}I_{\rm PAR},I_{\min}\right).$$

(29)

Grazing of zooplankton on phytoplankton

$$d_{pz}=g_{\max}\left(1-\exp \left(-I_{v}^{2}c_{p}^{2}\right)\right)c_{z} $$

(30)

Phytoplankton excretion

$$ d_{pn} = r_{pn}c_{p}$$

(31)

Zooplankton excretion

$$ d_{zn} = r_{zn} c_{z} $$

(32)

Remineralisation of detritus into nutrients

$$ d_{dn} = r_{dn} c_{d}$$

(33)

Phytoplankton mortality

$$ d_{pd} = r_{pd} c_{p}$$

(34)

Zooplankton mortality

$$ d_{zd} = r_{zd} c_{z}$$

(35)

All empirical parameters for the NPZD model are given in Table 4.

Table 4 Empirical constants used for the NPZD model