Analytic sources using polynomial shaped particles in the LTP method

The Local Taylor Polynomial (LTP) method treats fields and sources as local analytic polynomials for rapid numerical solution of PDE's. Knowledge of the source polynomial would allow sources to be treated analytically in LTP formulas, since the derivatives follow directly for polynomial forms. Aside from idealized cases, such information is seldom available; however, use of macro-particles with finite polynomial shape functions for charge and current deposition can make analytic treatment possible for general cases. The macro-particle polynomial coefficients can be used directly in LTP solution formulas. This scheme allows one to choose a representation of particles that enables high fidelity with fewer particles compared to traditional particle methods, since the polynomials can include information such as transverse profile information from a beam, for example. This paper presents a simple polynomial form for representing macro-particles in numerical simulations. The particle shape must be continuous and differentiable to a specified order, bounded in size and magnitude, zero in magnitude and derivatives at the edges, symmetric, positive definite, with analytic coefficients. Two key properties for use in LTP will be demonstrated: zero derivatives at the particle boundary and analytically computable coefficients.