Applied and Computational Mathematics (ACM)

Nonstandard finite difference schemes (NSFD)

Nonstandard finite difference (NSFD) methods are tailor made special

schemes for the numerical integration of differential equations in order to preserve certain properties (positivity, asymptotic behaviour, etc.) of the analytic solution on the discrete level.

Detailed studies of so-called "exact finite difference schemes" are the foundation of NSFD methods. The extension and generalization of these results to special groups of differential equations for which exact schemes are not available has also provided additional insight into the required structural properties of NSFD methods.

According to Mickens, these general basic rules are the following

  1. The orders of the discrete derivatives should be equal to the orders of the corresponding derivatives appearing in the differential equations.
  2. Discrete representations for derivatives must, in general, have nontrivial denominator functions.
  3. Nonlinear terms should, in general, be replaced by nonlocal discrete representations.
  4. Special conditions that hold for either the differential equation and/or its solutions should also hold for the difference equation model and/or its solutions.


  • Prof. Dr. Ronald E. Mickens, Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA.


  • M. Ehrhardt and R.E. Mickens, A Nonstandard finite difference scheme for the Black-Scholes equation of Option pricing, Preprint 12/08, March 2012.
  • M. Ehrhardt and R.E. Mickens, Discrete Models for the Cube-Root Differential Equation, Neural, Parallel, and Scientific Computations Vol. 16, Number 2, (2008), 179-188.


  • R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
  • R. E. Mickens (ed.), Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, 2000.

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