TY - JOUR
T1 - Random Walk Approximation for Irreversible Drift-Diffusion Process on Manifold: Ergodicity, Unconditional Stability and Convergence
AU - Gao , Yuan
AU - Liu , Jian-Guo
JO - Communications in Computational Physics
VL - 1
SP - 132
EP - 172
PY - 2023
DA - 2023/08
SN - 34
DO - http://doi.org/10.4208/cicp.OA-2023-0021
UR - https://global-sci.org/intro/article_detail/cicp/21883.html
KW - Symmetric decomposition, non-equilibrium thermodynamics, enhancement by mixture, exponential ergodicity, structure-preserving numerical scheme.
AB - <p style="text-align: justify;">Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does
not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed
manifold, using Voronoi tessellation, we propose two upwind finite volume schemes
with or without the information of the invariant measure. Both schemes possess
stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and
a Hamiltonian flow part, enabling us to prove unconditional stability, ergodicity and
error estimates. Based on the two upwind schemes, several numerical examples – including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system – are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. This makes them suitable for adapting
to manifold-related computations that arise from high-dimensional molecular dynamics simulations.</p>