Finite range method of approximation for balance laws in measure spaces

Piotr Gwiazda; Piotr Orlinski; Agnieszka Ulikowska

doi:10.3934/krm.2017027

Article Contents

2017, Volume 10, Issue 3: 669-688. Doi: 10.3934/krm.2017027

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Finite range method of approximation for balance laws in measure spaces

Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Piotr Gwiazda
* Corresponding author: Piotr Gwiazda

Received: March 31, 2016

Revised: June 30, 2016

Published: September 2017

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Abstract

In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

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Mathematics Subject Classification: P92D25, 65M12, 65M75.

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Figure 1. Function $f$ and its Finite Range Approximation

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Figure 2. Function $f^{\varepsilon}$ and its approximation $\bar f^{\varepsilon}$

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Figure 3. Order of convergence

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Table 1. The error of the FRA method for $\varepsilon=0.1$

$\Delta t$	${\rm{Err}}(1,\Delta t,0.1)$	$\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$	$0.0008865973$	-
$5.0000\cdot 10^{-2}$	$0.0005886514$	0.59086543
$2.5000\cdot 10^{-2}$	$0.0003434678$	0.77723882
$1.2500\cdot 10^{-2}$	$0.0002668756$	0.36400752
$6.2500\cdot 10^{-3}$	$0.0002400045$	0.15310595
$3.1250\cdot 10^{-3}$	$0.0002258416$	0.08775007
$1.5625\cdot 10^{-3}$	$0.0002187803$	0.04582869
$7.8125\cdot 10^{-4}$	$0.0002154824$	0.02191237

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Table 2. The error of the FRA method for $\varepsilon=0.0125$

$\Delta t$	${\rm{Err}}(1,\Delta t,0.0125)$	$\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$
$1.0000\cdot 10^{-1}$	$6.702793\cdot 10^{-4}$	-
$5.0000\cdot 10^{-2}$	$4.255311\cdot 10^{-4}$	0.6554979
$2.5000\cdot 10^{-2}$	$1.757573\cdot 10^{-4}$	1.2756799
$1.2500\cdot 10^{-2}$	$9.330613\cdot 10^{-5}$	0.9135408
$6.2500\cdot 10^{-3}$	$5.658711\cdot 10^{-5}$	0.7214983
$3.1250\cdot 10^{-3}$	$4.055238\cdot 10^{-5}$	0.4806869
$1.5625\cdot 10^{-3}$	$3.332588\cdot 10^{-5}$	0.2831438
$7.8125\cdot 10^{-4}$	$2.973577\cdot 10^{-5}$	0.1644434

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Table 3. The error of the FRA method for $\varepsilon=0.0015625$

$\Delta t$	${\rm{Err}}(1,\Delta t,0.0015625)$	$\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$	$6.616854\cdot 10^{-4}$	-
$5.0000\cdot 10^{-2}$	$4.139056\cdot 10^{-4}$	0.6768439
$2.5000\cdot 10^{-2}$	$1.570582\cdot 10^{-4}$	1.3980025
$1.2500\cdot 10^{-2}$	$7.418803\cdot 10^{-5}$	1.0820407
$6.2500\cdot 10^{-3}$	$3.649949\cdot 10^{-5}$	1.0820407
$3.1250\cdot 10^{-3}$	$1.883518\cdot 10^{-5}$	0.9544464
$1.5625\cdot 10^{-3}$	$1.032763\cdot 10^{-5}$	0.8669200
$7.8125\cdot 10^{-4}$	$6.277910\cdot 10^{-6}$	0.7181537

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Table 4. The error of the FRA method for $\varepsilon=\Delta t$

$\Delta t$	${\rm{Err}}(1,\Delta t,\Delta t)$	$\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$
$1.0000\cdot 10^{-1}$	$8.865973\cdot 10^{-4}$	-
$5.0000\cdot 10^{-2}$	$4.797352\cdot 10^{-4}$	0.8860407
$2.5000\cdot 10^{-2}$	$1.991736\cdot 10^{-4}$	1.2682122
$1.2500\cdot 10^{-2}$	$9.330613\cdot 10^{-5}$	1.0939823
$6.2500\cdot 10^{-3}$	$4.477800\cdot 10^{-5}$	1.0591816
$3.1250\cdot 10^{-3}$	$2.162411\cdot 10^{-5}$	1.0501496
$1.5625\cdot 10^{-3}$	$1.032763\cdot 10^{-5}$	1.0661308
$7.8125\cdot 10^{-4}$	$4.763338\cdot 10^{-6}$	1.1164650

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