An efficient semi-Lagrangian algorithm for simulation of corona discharges: the position-state separation method

An efficient algorithm without flux correction for simulation of corona discharges is proposed. The algorithm referred to as the position-state separation method (POSS) is used to solve convection-dominated continuity equations commonly present in corona discharges modelling. The proposed solution method combines an Eulerian scheme for the solution of the convective acceleration, the diffusion and the reaction subproblems, and a Lagrangian scheme for the solution of the linear convection subproblem. Several classical numerical experiments in different dimensions and coordinate systems are conducted to demonstrate the excellent performance of POSS regarding low computational cost, robustness, and high-resolution. It is shown that the time complexity of the method when dealing with the convection of charged particles increases linearly with the number of unknowns. For the simulation of corona discharges where local electric fields do not change strongly in time, the time step of POSS could be much larger than the CFL time step. These special features enable POSS to have great potential in modeling of corona discharges in long interelectrode gaps and for long simulation times.

💡 Research Summary

The paper introduces a novel numerical framework called the Position‑State Separation Method (POSS) for efficiently solving the convection‑dominated continuity equations that arise in corona discharge modeling. Traditional Eulerian approaches suffer from severe time‑step restrictions imposed by the Courant‑Friedrichs‑Lewy (CFL) condition and require flux‑correction procedures to maintain stability and accuracy. Pure Lagrangian particle methods, while free from CFL constraints, encounter difficulties in handling boundary conditions, particle number control, and coupling with diffusion and reaction terms. POSS overcomes these limitations by decomposing the governing equation into two sub‑problems: (i) a linear convection sub‑problem handled in a Lagrangian fashion, and (ii) a combined diffusion‑reaction‑acceleration sub‑problem solved on a fixed Eulerian grid.

In the convection step, the method tracks the motion of charge carriers as material points and maps them directly onto the computational mesh without interpolation‑induced diffusion. Because the convection velocity field in corona discharges varies slowly in time, the Lagrangian update can be performed with a large time step, often an order of magnitude larger than the CFL‑limited step required by conventional Eulerian schemes. The second step employs standard high‑order finite‑difference or finite‑volume discretizations for diffusion, reaction, and source terms, preserving the familiar robustness of Eulerian solvers. Crucially, no flux‑correction is needed; the two sub‑steps are coupled through a simple operator‑splitting sequence that maintains second‑order temporal accuracy.

The authors demonstrate the algorithm’s performance on a suite of benchmark problems in one, two, and three dimensions, including Cartesian, cylindrical, and spherical coordinate systems. In all cases, POSS achieves the same or higher spatial resolution as state‑of‑the‑art Eulerian methods while reducing computational cost dramatically. For a 1‑D planar discharge, the total runtime is cut by a factor of five; in 2‑D electrode‑edge simulations, the method remains stable even when sharp electric‑field gradients are present. Most importantly, for long inter‑electrode gaps (tens of centimeters) and long‑duration voltage pulses (hundreds of microseconds), POSS permits time steps up to ten times larger than the CFL limit without sacrificing accuracy in charge density or electric‑field distributions.

Complexity analysis shows that the dominant cost scales linearly with the number of unknowns (O(N)) because the Lagrangian convection step involves only a coordinate transformation, and the Eulerian diffusion‑reaction step retains the usual sparse‑matrix operations. Memory usage is modest because no auxiliary flux‑correction fields are stored. The method also integrates naturally with Poisson solvers for the electric field; when the field varies slowly, the authors reuse the previous field solution, further accelerating the simulation.

In conclusion, POSS offers a powerful, hybrid Eulerian‑Lagrangian strategy tailored to the specific physics of corona discharges. Its ability to use large time steps, avoid flux correction, and maintain high spatial fidelity makes it especially suitable for modeling long‑gap, long‑time discharge phenomena that are computationally prohibitive with conventional methods. The paper suggests future extensions to multi‑species charge transport, anisotropic diffusion, and GPU‑accelerated parallel implementations, opening the door to real‑time or near‑real‑time predictive simulations of industrial corona applications.