Dual-Mesh Characteristics for Particle-Mesh Methods for the Simulation of Convection-Dominated Flows

Abstract

The particle-mesh method (PMM) is a powerful computational tool for the simulation of convection-dominated diffusion flows. The method introduces computational particles each of which is given a finite size and represents a large number of physical particles with the same properties. The convection part of the flow can be solved by moving the computational particles along the characteristics, while the diffusion part is carried out by utilizing a heat solver on a regular mesh. However, the method in practical applications shows the so-called ringing instability, an amplitude fluctuation in the computed solution. In this article, we suggest a new numerical technique of particle movement, called the dual-mesh characteristics (DMC) of which the second mesh is formed by tracking back the cells along the characteristics. The particle movement is carried out by interpreting the particle positions (in the previous time level) in terms of the multi-linear coordinates of the second mesh. Strategies for the average velocity and interpolations are also suggested to move the computational particles accurately with a minimum numerical dissipation. The resulting algorithm, PMM-DMC, turns out to be mass-conservative, non-oscillatory, of negligible dissipation, and more efficient than the conventional schemes. Numerical results are shown to demonstrate its accuracy and efficiency.

Fluid Simulation on Neural Flow Maps

“…The Stable Fluids method [Stam 1999] yields a significant amount of numerical viscosity causing the results to appear blurred and damped. Researchers have addressed this artifact with error correction schemes [Kim et al 2006;Selle et al 2008], higher-order interpolation [Losasso et al 2006b;Nave et al 2009], improved backtracking schemes [Cho et al 2018;Jameson et al 1981], particle-based advection [Fu et al 2017;Jiang et al 2015;Zhu and Bridson 2005], energy-preserving integration, [Mullen et al 2009], a posteriori vorticity correction [Fedkiw et al 2001;, reflection [Narain et al 2019;Strang 1968;Zehnder et al 2018] and Lie advection [Nabizadeh et al 2022]. Flow map methods offer another alternative option, as elaborated below.…”

“…a semi-Lagrangian-based scheme is used for solving 𝜓 (we use the dual-mesh characteristics [Cho et al 2018] for backtracking as suggested by Qu et al [2019]), and the 4 th order Runge-Kutta scheme for solving 𝜙. Both T and F are computed from 𝜓 and 𝜙 using finite difference.…”

Efficient and conservative fluids using bidirectional mapping

“…Unfortunately, in highly rotational velocity fields, this simple strategy drastically loses its accuracy. To alleviate these issues, we apply dual-mesh characteristic from [Cho et al 2018], which improves the numerical accuracy for the mapping advection.…”