Integrated Greens Function implementation for 2D, 2.5D, slice-by-slice, and 3D FFT-based Poisson solvers

[woodtp]

Adds optional integrated Green’s Function (IGF) kernels to the 2D and 3D FFT Poisson solvers, following the implementation as described in [1].

The existing point-sampled kernel treats the charge deposited at each grid point as a point source. This method instead integrates the free-space Green’s function over the finite source cell, which may improve accuracy when the grid cells' aspect ratios are large.

Qiang et al. propose four strategies for mitigating cancellation in terms such as $\log(x+r)$. When $x<0$ and $\lvert x\rvert\approx r$, direct evaluation of $x+r$ subtracts two nearly equal numbers and can become limited by numerical precision. The same issue applies independently to the $y+r$ and $z+r$ terms.

I chose the algebraic rewrite from Equation (11). From the included benchmark, the resulting maximum normalized field error remained below 0.1%, including the case where direct evaluation of the logarithm failed.

double stableLogArgument(double coordinate, double other_a, double other_b, double radius)
{
	if (coordinate >= 0.0) {
		return std::log(coordinate + radius);
	}
	return std::log((other_a * other_a + other_b * other_b) / (radius - coordinate));
}

• Added GreensFunction2D and GreensFunction3D helpers for finite-cell Green’s-function evaluation.
• Added setUseIntegratedGreenFunction(bool) and getUseIntegratedGreenFunction() to PoissonSolver* and PoissonSolverFFT3D.
• Builds the selected kernel directly into the solver’s FFTW kernel buffer; removes the obsolete intermediate double** double*** kernel storage.
• Exposed the option through Python for:
  • PoissonSolverFFT2D
  • PoissonSolverFFT3D
  • SpaceChargeCalc3D
  • SpaceChargeCalc2p5D
  • SpaceChargeCalc2p5Drb
  • SpaceChargeCalcSliceBySlice2D

[1] Qiang, J., Mitchell, C., Lehe, R., & Formenti, A. (2024). Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain. Journal of Software Engineering and Applications, 17(9), 740–749. https://doi.org/10.4236/jsea.2024.179039