Improved Space Charge Modeling for Simulation and Design of Photoinjectors Page: 4 of 36
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Calabazas Creek Research Topic 38g: Improved Space Charge Modeling for Simulation and Design of Photoinjectors
r
z
Figure 1: Curvature and special grid nodes in cylindrical coordinates. Note that there are three distinct
node types: 1) axis nodes (red), 2) first-row nodes (blue) and 3) "standard" nodes. To obtain the highest
solution accuracies the axis and first row nodes require separate treatment.
2 The Local Taylor Polynomial Method
The Local Taylor Polynomial (LTP) method is similar to standard finite differences in that both
can be related to a Taylor expansion of the solution function around a point. One key difference
is that LTP is based on a local analytic solution (polynomial in this initial work) of the
continuum partial differential equation (PDE). Therefore, the polynomial satisfies the PDE at
each order but is not necessarily an exact solution because it has to be truncated at some finite
order. Discrete numerical formulas are then developed using this analytic solution polynomial. In
contrast, finite difference formulas enforce the "solution" through discrete approximation of the
PDE. One key benefit of using a polynomial rather than other local representations is the
simplicity of taking derivatives. However, other representations may require fewer terms to
achieve equivalent fidelity, to be explored in future work.
An important benefit of the LTP approach is that the source function becomes a full component
in the formulation, on an equal footing with the field function. This provides a consistent
methodology for addressing source effects in a variety of situations, e.g. near symmetry
boundaries such as the cylindrical axis, or near PEC or dielectric boundaries. This capability is
lacking in standard finite difference developments.
We begin our exposition of the LTP technique using the 1-D scalar Poisson PDE in cylindrical
coordinates. This 1-D example serves to highlight features of the LTP method without the
distracting details of higher dimensions. Also, we will restrict the discussion to the Poisson PDE,
but note that the method is quite general and applicable to other PDE's.
Following this introduction, a general matrix methodology for implementing the LTP method
will be developed and discussed. Examples will be given for 2-D cylindrical coordinates and
some special cases will be addressed, e.g. on-axis and first-row formulas, non-conformal
boundaries, etc. Application of the LTP to electric field calculation is then examined.
2.1 The Local Taylor Polynomial
LTP starts with general polynomials for the field and source functions written in terms of
"offset" coordinates (the "local" in LTP) as shown in the expressions below. Here rp is the4
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Robert H. Jackson, Thuc Bui, John Verboncoeur. Improved Space Charge Modeling for Simulation and Design of Photoinjectors, report, April 19, 2010; United States. (https://digital.library.unt.edu/ark:/67531/metadc841927/m1/4/: accessed March 28, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.