33 Field Solver Physics
This chapter is the physics-level counterpart to the Field Solver user guide. The user guide documents input choices and current runtime behavior; this chapter will derive the model, state its approximations, and collect numerical validation.
The headings below define the intended scope. They deliberately contain only short writing prompts so that equations and claims can be added after they are checked against the implementation and benchmark cases.
33.1 Scope and notation
Define the charge, field, coordinate, unit, and sign conventions used throughout the chapter. State which parts apply to all solvers and which are specific to space charge.
33.2 Governing electrostatic model
State the quasi-static approximation and the physical problem solved in the selected bunch frame.
33.2.1 Poisson equation
Introduce the source term, potential, electric field, and boundary data.
33.2.2 Self-field force
Connect the solved fields to the force applied by the tracker, including any frame transformation required before the particle push.
33.3 Particle-mesh discretization
Describe the complete particle-in-cell cycle and identify the discrete quantities that live on particles and on the mesh.
33.3.1 Charge deposition
Specify the particle shape, weighting rule, normalization, and treatment of ghost cells.
33.3.2 Mesh solve
Define the discrete Poisson operator or convolution used by each solver family.
33.3.3 Field reconstruction and interpolation
Explain how the electric field is reconstructed and interpolated back to particle positions.
33.4 Frames and relativistic transformations
Derive the transformations between the laboratory frame, mean rest frame, and bin-local frames. State where approximations enter.
33.5 Boundary conditions and Green functions
Define each boundary-value problem independently of its input syntax.
33.5.1 Periodic boundaries
Describe the periodic domain, compatibility conditions, and zero-mode handling.
33.5.2 Open boundaries
Derive the free-space Green-function convolution and doubled-domain method.
33.5.3 Dirichlet and image-charge boundaries
Explain explicit image-charge and shifted-Green constructions, including the geometries and assumptions for which they are valid.
33.6 Solver formulations
Relate the mathematical problem to the available numerical backends.
33.6.1 Periodic FFT solver
Document the spectral formulation, differentiation, normalization, and parallel decomposition.
33.6.2 Hockney open-boundary solver
Document domain doubling, kernel sampling, padding, and extraction of the physical-domain result.
33.6.3 Iterative solvers
Reserve this subsection for the operator, convergence criterion, preconditioning, and supported boundary conditions once an iterative backend is production-ready.
33.7 Binned rest-frame space charge
Derive the longitudinal binning approximation, bin-frame transforms, per-bin solve, and accumulation of the resulting fields.
33.8 Emission and conducting boundaries
Describe how emission time, cathode geometry, image charges, and activation of space charge interact.
33.9 Numerical accuracy and convergence
Collect the error sources that should be varied in convergence studies.
33.9.1 Mesh resolution
Define spatial refinement studies and expected convergence observables.
33.9.2 Particle noise and deposition order
Separate sampling noise from mesh and solver error.
33.9.3 Time-step coupling
Explain how field-solve frequency, particle stepping, and emission updates couple to the spatial solve.
33.10 Domain decomposition and load balancing
Explain which mathematical data are repartitioned and why the field, particle, and solver state must be refreshed in a defined order.
33.11 Current implementation architecture
The following diagram is retained as TikZ source for PDF output. HTML uses a small generated PNG of the same source so the page does not depend on a browser-side TikZ renderer.
figures/current-space-charge-class-diagram.tex.
33.12 Verification and benchmarks
List analytic solutions, manufactured solutions, regression cases, conserved quantities, and cross-code comparisons required for each solver mode.
33.13 Assumptions and known limitations
Maintain a concise list of model assumptions, unsupported combinations, and known numerical limitations, with links to reproducible investigations.