Stability and errors estimates of a second-order IMSP scheme

Fasma Diele; Angela Martiradonna; Catalin Trenchea

doi:10.3934/dcdss.2022076

Article Contents

2022, Volume 15, Issue 12: 3645-3665. Doi: 10.3934/dcdss.2022076

This issue Previous Article Transmission problems for the fractional $ p $-Laplacian across fractal interfaces Next Article Spline based Hermite quasi-interpolation for univariate time series

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
^* Corresponding author: angela.martiradonna@uniba.it

Received: September 2021

Revised: January 2022

Early access: March 2022

Published: December 2022

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 65M60, 65M15, 65M12, 35K57, 35K55.

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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

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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

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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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