# American Institute of Mathematical Sciences

September  2017, 10(3): 669-688. doi: 10.3934/krm.2017027

## 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

Received  April 2016 Revised  July 2016 Published  December 2016

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.

Citation: Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
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##### References:
Function $f$ and its Finite Range Approximation
Function $f^{\varepsilon}$ and its approximation $\bar f^{\varepsilon}$
Order of convergence

Results for $\varepsilon \in \{0.1, 0.0125, 0.0015625, \Delta t\}$ presented in Tables 1-4

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
 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.1)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $0.0008865973$ - $5.0000\cdot 10^{-2}$ $0.0005886514$ 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
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
 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.0125)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}})/ \log 2$ $1.0000\cdot 10^{-1}$ $6.702793\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.255311\cdot 10^{-4}$ 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
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
 $\Delta t$ ${\rm{Err}}(1,\Delta t,0.0015625)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $6.616854\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.139056\cdot 10^{-4}$ 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
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
 $\Delta t$ ${\rm{Err}}(1,\Delta t,\Delta t)$ $\log({\frac{{\rm{Err}}(T,2\Delta t,2\varepsilon)}{{\rm{Err}}(T,\Delta t,\varepsilon)}}) / \log 2$ $1.0000\cdot 10^{-1}$ $8.865973\cdot 10^{-4}$ - $5.0000\cdot 10^{-2}$ $4.797352\cdot 10^{-4}$ 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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