Finite Range Method of Approximation for Balance Laws in Measure Spaces

TL;DR

This paper introduces a modified numerical scheme for measure-valued solutions of balance laws, improving computational efficiency by controlling the growth of Dirac measures while maintaining convergence properties.

Contribution

It proposes a finite range approximation technique that prevents exponential growth in Dirac measures, enhancing the existing kinetic-based scheme for population models.

Findings

The modified scheme converges with proven speed.

Numerical simulations demonstrate effectiveness across test cases.

The approach improves computational efficiency in measure space approximations.

Abstract

In the following paper we reconsider a recently introduced numerical scheme. The method was designed for a wide class of size structured population models as a variation of the Escalator Boxcar Train (EBT) method, which is commonly used in computational biology. The scheme under consideration bases on the kinetic approach and the split-up technique - it approximates a solution by a sum of Dirac measures at each discrete time moment. In the current paper we propose a modification of this algorithm, which prevents (possible) exponential growth of the number of Dirac Deltas approximating the solution. Our approach bases on the finite range approximation of a coefficient which describes birth processes in a population. We provide convergence results, including the convergence speed. Moreover, some results of numerical simulations for several test cases are shown.