Non-Taylor Discretization

Abstract

One has to frequently deal with systems described by differential equations, which have to be solved subject to prescribed boundary and initial conditions. The first step in numerical solution is the replacement of derivatives with finite difference approximations, which are normally derived via Taylor's series. Here, superior finite difference approximations are obtained by non-Taylor methods. For this purpose, the shift operator in the Taylor's series is expressed as an exponential function in the differential operator and that function is expanded using approximation theoretic notions involving Chebyshev or Bernstein polynomials, among other possibilities. These non-Taylor finite difference approximations lead to new finite difference schemes without increasing their complexity.

Indexing terms:

• Chebyshev method
• Finite difference simulation
• Approximation theory
• Bernstein polynomials