An asymptotic preserving scheme for kinetic models with singular limit
This work addresses the challenge of numerically solving kinetic models with singular limits, which is important for applications in biology and physics.
The paper proposes a new asymptotic preserving scheme for kinetic equations with a mono-kinetic singular limit, demonstrating accuracy, stability, and efficiency through numerical experiments on two biologically related systems.
We propose a new class of asymptotic preserving schemes to solve kinetic equations with mono-kinetic singular limit. The main idea to deal with the singularity is to transform the equations by appropriate scalings in velocity. In particular, we study two biologically related kinetic systems. We derive the scaling factors and prove that the rescaled solution does not have a singular limit, under appropriate spatial non-oscillatory assumptions, which can be verified numerically by a newly developed asymptotic preserving scheme. We set up a few numerical experiments to demonstrate the accuracy, stability, efficiency and asymptotic preserving property of the schemes.