doi: 10.3934/dcdss.2022076
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Stability and errors estimates of a second-order IMSP scheme

1. 

Istituto per le applicazioni del Calcolo M. Picone, CNR - Bari, via Amendola 122/D, Italy

2. 

Department of Mathematics, University of Bari Aldo Moro, via Orabona 4, Bari, Italy

3. 

Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, PA15260, USA

* Corresponding author: angela.martiradonna@uniba.it

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: Partially supported by the AFOSR under grant FA 9550-16-1-0355 and the NSF under grantDMS 1522574. This work has received fundings from the REFIN project N.0C46E06B, Regione Puglia, Italy

We analyze a second-order accurate implicit-symplectic (IMSP) scheme for reaction-diffusion systems modeling spatiotemporal dynamics of predator-prey populations. We prove stability and errors estimates of the semi-discrete-in-time approximations, under positivity assumptions. The numerical simulations confirm the theoretically derived rates of convergence and show an improved accuracy in the second-order IMSP in comparison with the first-order IMSP, at same computational cost.

Citation: Fasma Diele, Angela Martiradonna, Catalin Trenchea. Stability and errors estimates of a second-order IMSP scheme. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022076
References:
[1]

M. BukacW. LaytonM. MoraitiH. Tran and C. Trenchea, Analysis of partitioned methods for the biot system, Numerical Methods for Partial Differential Equations, 31 (2015), 1769-1813.  doi: 10.1002/num.21968.

[2]

M. Bukač, A. Seboldt and C. Trenchea, Refactorization of Cauchy's method: A second-order partitioned method for fluid-thick structure interaction problems, J. Math. Fluid Mech., 23 (2021), Paper No. 64, 25 pp. doi: 10.1007/s00021-021-00593-z.

[3]

J. Burkardt and C. Trenchea, Refactorization of the midpoint rule, Appl. Math. Lett., 107 (2020), 106438, 7 pp. doi: 10.1016/j.aml.2020.106438.

[4]

F. DieleM. Garvie and C. Trenchea, Numerical analysis of a first-order in time implicit-symplectic scheme for predator–prey systems, Comput. Math. Appl., 74 (2017), 948-961.  doi: 10.1016/j.camwa.2017.04.030.

[5]

F. Diele and C. Marangi, Positive symplectic integrators for predator-prey dynamics, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2661-2678.  doi: 10.3934/dcdsb.2017185.

[6]

F. Diele and C. Marangi, Geometric numerical integration in ecological modelling, Mathematics, 8 (2020), 25.  doi: 10.3390/math8010025.

[7]

F. DieleC. Marangi and S. Ragni, Implicit - symplectic partitioned (IMSP) Runge-Kutta schemes for predator-prey dynamics, AIP Conference Proceedings, 1479 (2012), 1177-1180.  doi: 10.1063/1.4756360.

[8]

F. DieleC. Marangi and S. Ragni, IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics, Math. Comput. Simulation, 100 (2014), 41-53.  doi: 10.1016/j.matcom.2014.01.006.

[9]

V. ErvinM. KubackiW. LaytonM. MoraitiZ. Si and C. Trenchea, Partitioned penalty methods for the transport equation in the evolutionary Stokes-Darcy-transport problem, Numer. Methods Partial Differential Equations, 35 (2019), 349-374.  doi: 10.1002/num.22303.

[10]

M. R. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull. Math. Biol., 69 (2007), 931-956.  doi: 10.1007/s11538-006-9062-3.

[11]

M. R. GarvieJ. Burkardt and J. Morgan, Simple finite element methods for approximating predator-prey dynamics in two dimensions using Matlab, Bull. Math. Biol., 77 (2015), 548-578.  doi: 10.1007/s11538-015-0062-z.

[12]

M. R. Garvie and C. Trenchea, Spatiotemporal dynamics of two generic predator-prey models, J. Biol. Dyn., 4 (2010), 559-570.  doi: 10.1080/17513750903484321.

[13]

M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type Ⅱ functional response, Numer. Math., 107 (2007), 641-667.  doi: 10.1007/s00211-007-0106-x.

[14]

W. Gentleman, A. Leising, B. Frost, S. Strom and J. Murray, Functional responses for zooplankton feeding on multiple resources: A review of assumptions and biological dynamics, Deep Sea Research Part Ⅱ: Topical Studies in Oceanography, 50 (2003), 2847–2875, http://www.sciencedirect.com/science/article/pii/S0967064503001711. doi: 10.1016/j.dsr2.2003.07.001.

[15]

G. GuidoboniR. GlowinskiN. Cavallini and S. Canic, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937.  doi: 10.1016/j.jcp.2009.06.007.

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Reprint of the second (2006) edition, Springer Series in Computational Mathematics, 31. Springer, Heidelberg, 2010.

[17]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Ⅳ. Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[18]

C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385–398, http://journals.cambridge.org/article-S0008347X00072692. doi: 10.4039/Ent91385-7.

[19]

W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, 33. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-09017-6.

[20]

J. M. Jeschke, M. Kopp and R. Tollrian, Predator functional responses: Discriminating between handling and digesting prey, Ecological Monographs, 72 (2002), 95–112, http://www.jstor.org/stable/3100087.

[21]

T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, J. Comput. Appl. Math., 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.

[22]

Y. Li and C. Trenchea, Partitioned second order method for magnetohydrodynamics in Elsässer variables, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2803-2823.  doi: 10.3934/dcdsb.2018106.

[23]

H. MalchowS. Petrovskii and A. Medvinsky, Numerical study of plankton-fish dynamics in a spatially structured and noisy environment, Ecol. Model., 149 (2002), 247-255.  doi: 10.1016/S0304-3800(01)00467-7.

[24]

G. Marinoschi and A. Martiradonna, Fish populations dynamics with nonlinear stock-recruitment renewal conditions, Applied Mathematics and Computation, 277 (2016), 101-110.  doi: 10.1016/j.amc.2015.12.041.

[25]

A. MartiradonnaG. Colonna and F. Diele, GeCo: Geometric Conservative nonstandard schemes for biochemical systems, Applied Numerical Mathematics, 155 (2020), 38-57.  doi: 10.1016/j.apnum.2019.12.004.

[26]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[27]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209–223, http://www.jstor.org/stable/2458702. doi: 10.1086/282272.

[28]

G. Settanni and I. Sgura, Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems, Journal of Computational and Applied Mathematics, 292 (2016), 674-693.  doi: 10.1016/j.cam.2015.04.044.

[29]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type Ⅱ model, Ecology, 82 (2001), 3083-3092. 

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.

[31]

C. Trenchea, Partitioned conservative, variable step, second-order method for magneto-hydrodynamics in Elsässer variables, ROMAI J., 15 (2019), 117-137. 

show all references

References:
[1]

M. BukacW. LaytonM. MoraitiH. Tran and C. Trenchea, Analysis of partitioned methods for the biot system, Numerical Methods for Partial Differential Equations, 31 (2015), 1769-1813.  doi: 10.1002/num.21968.

[2]

M. Bukač, A. Seboldt and C. Trenchea, Refactorization of Cauchy's method: A second-order partitioned method for fluid-thick structure interaction problems, J. Math. Fluid Mech., 23 (2021), Paper No. 64, 25 pp. doi: 10.1007/s00021-021-00593-z.

[3]

J. Burkardt and C. Trenchea, Refactorization of the midpoint rule, Appl. Math. Lett., 107 (2020), 106438, 7 pp. doi: 10.1016/j.aml.2020.106438.

[4]

F. DieleM. Garvie and C. Trenchea, Numerical analysis of a first-order in time implicit-symplectic scheme for predator–prey systems, Comput. Math. Appl., 74 (2017), 948-961.  doi: 10.1016/j.camwa.2017.04.030.

[5]

F. Diele and C. Marangi, Positive symplectic integrators for predator-prey dynamics, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2661-2678.  doi: 10.3934/dcdsb.2017185.

[6]

F. Diele and C. Marangi, Geometric numerical integration in ecological modelling, Mathematics, 8 (2020), 25.  doi: 10.3390/math8010025.

[7]

F. DieleC. Marangi and S. Ragni, Implicit - symplectic partitioned (IMSP) Runge-Kutta schemes for predator-prey dynamics, AIP Conference Proceedings, 1479 (2012), 1177-1180.  doi: 10.1063/1.4756360.

[8]

F. DieleC. Marangi and S. Ragni, IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics, Math. Comput. Simulation, 100 (2014), 41-53.  doi: 10.1016/j.matcom.2014.01.006.

[9]

V. ErvinM. KubackiW. LaytonM. MoraitiZ. Si and C. Trenchea, Partitioned penalty methods for the transport equation in the evolutionary Stokes-Darcy-transport problem, Numer. Methods Partial Differential Equations, 35 (2019), 349-374.  doi: 10.1002/num.22303.

[10]

M. R. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull. Math. Biol., 69 (2007), 931-956.  doi: 10.1007/s11538-006-9062-3.

[11]

M. R. GarvieJ. Burkardt and J. Morgan, Simple finite element methods for approximating predator-prey dynamics in two dimensions using Matlab, Bull. Math. Biol., 77 (2015), 548-578.  doi: 10.1007/s11538-015-0062-z.

[12]

M. R. Garvie and C. Trenchea, Spatiotemporal dynamics of two generic predator-prey models, J. Biol. Dyn., 4 (2010), 559-570.  doi: 10.1080/17513750903484321.

[13]

M. R. Garvie and C. Trenchea, Finite element approximation of spatially extended predator-prey interactions with the Holling type Ⅱ functional response, Numer. Math., 107 (2007), 641-667.  doi: 10.1007/s00211-007-0106-x.

[14]

W. Gentleman, A. Leising, B. Frost, S. Strom and J. Murray, Functional responses for zooplankton feeding on multiple resources: A review of assumptions and biological dynamics, Deep Sea Research Part Ⅱ: Topical Studies in Oceanography, 50 (2003), 2847–2875, http://www.sciencedirect.com/science/article/pii/S0967064503001711. doi: 10.1016/j.dsr2.2003.07.001.

[15]

G. GuidoboniR. GlowinskiN. Cavallini and S. Canic, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, J. Comput. Phys., 228 (2009), 6916-6937.  doi: 10.1016/j.jcp.2009.06.007.

[16]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Reprint of the second (2006) edition, Springer Series in Computational Mathematics, 31. Springer, Heidelberg, 2010.

[17]

J. G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. Ⅳ. Error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384.  doi: 10.1137/0727022.

[18]

C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385–398, http://journals.cambridge.org/article-S0008347X00072692. doi: 10.4039/Ent91385-7.

[19]

W. Hundsdorfer and J. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, 33. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-09017-6.

[20]

J. M. Jeschke, M. Kopp and R. Tollrian, Predator functional responses: Discriminating between handling and digesting prey, Ecological Monographs, 72 (2002), 95–112, http://www.jstor.org/stable/3100087.

[21]

T. Koto, IMEX Runge-Kutta schemes for reaction-diffusion equations, J. Comput. Appl. Math., 215 (2008), 182-195.  doi: 10.1016/j.cam.2007.04.003.

[22]

Y. Li and C. Trenchea, Partitioned second order method for magnetohydrodynamics in Elsässer variables, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2803-2823.  doi: 10.3934/dcdsb.2018106.

[23]

H. MalchowS. Petrovskii and A. Medvinsky, Numerical study of plankton-fish dynamics in a spatially structured and noisy environment, Ecol. Model., 149 (2002), 247-255.  doi: 10.1016/S0304-3800(01)00467-7.

[24]

G. Marinoschi and A. Martiradonna, Fish populations dynamics with nonlinear stock-recruitment renewal conditions, Applied Mathematics and Computation, 277 (2016), 101-110.  doi: 10.1016/j.amc.2015.12.041.

[25]

A. MartiradonnaG. Colonna and F. Diele, GeCo: Geometric Conservative nonstandard schemes for biochemical systems, Applied Numerical Mathematics, 155 (2020), 38-57.  doi: 10.1016/j.apnum.2019.12.004.

[26]

A. B. MedvinskyS. V. PetrovskiiI. A. TikhonovaH. Malchow and B.-L. Li, Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44 (2002), 311-370.  doi: 10.1137/S0036144502404442.

[27]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209–223, http://www.jstor.org/stable/2458702. doi: 10.1086/282272.

[28]

G. Settanni and I. Sgura, Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems, Journal of Computational and Applied Mathematics, 292 (2016), 674-693.  doi: 10.1016/j.cam.2015.04.044.

[29]

G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the holling type Ⅱ model, Ecology, 82 (2001), 3083-3092. 

[30]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Grundlehren der Mathematischen Wissenschaften, 258. Springer-Verlag, New York-Berlin, 1983.

[31]

C. Trenchea, Partitioned conservative, variable step, second-order method for magneto-hydrodynamics in Elsässer variables, ROMAI J., 15 (2019), 117-137. 

Figure 1.  On the left: Convergence rate and accuracy comparison between first- and second-order IMSPs schemes at $ T = 20 $. The two curves have different slopes, suggesting the different orders of convergence. The slope of the IMSP$ 1 $ curve is approximately $ 1 $, validating the first-order accuracy, while IMSP$ 2 $ slope is approximately $ 2 $, verifying the second-order accuracy. On the right: comparison between numerical errors and cputime for the first- and second-order IMSP schemes
Figure 2.  On the left: Convergence rate and accuracy comparison between first and second order IMSPs schemes at $ T = 50 $. The two curves have different slopes, this confirming the different order of convergence; the slope of IMSP$ 1 $ curve is approximately $ 1 $ denoting a first order accuracy while IMSP$ 2 $ slope is approximately equal to $ 2 $ denoting a second order accuracy. On the right: comparison between numerical error and cputime for first and second order IMSP schemes
Figure 3.  RM model (58)-(59). Spatial distribution of prey densities in the domain: on the left column IMSP first order approximations with $ \Delta t = 1/3,\, 1/24,\, 1/384 $, on the right column the approximation with IMSP second order scheme in correspondence of the same temporal stepsizes. Parameters: $ D_u = D_v = 1 $, $ \alpha = 0.4 $, $ \beta = 0.2 $, $ \gamma = 0.6 $. Initial conditions: $ u_0 = 6/35- 2 \cdot 10^{-7}( x - 0.1\cdot y - 225)( x - 0.1 \cdot y - 675) $ and $ v_0 = 116 / 245 - 3 \cdot 10^{-5}\cdot( x - 450 ) - 1.2 \cdot 10^{-4}( y - 150) $. Notice that IMSP2 approximation reaches convergence more quickly than IMSP1 scheme
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