Finite-Difference Method (FDM) Algorithm

Example(s):
Counter-Example(s):
See: Partial Differential Equation, Mathematics, Numerical Methods, Differential Equations, Recurrence Relation, Derivative, Discretization.

References

(Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Finite_difference_method Retrieved:2020-10-25.
- In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.
  Finite difference methods convert ordinary differential equations (ODE) or partial differential equations (PDE), which may be nonlinear, into a system of linear equations that can be solved by matrix algebra techniques. Modern computers can perform these linear algebra computations efficiently which, along with their relative ease of implementation, has led to the widespread use of FDM in modern numerical analysis^[1].
  Today, FDM are one of the most common approaches to the numerical solution of PDE, along with finite element methods .

↑ Christian Grossmann; Hans-G. Roos; Martin Stynes (2007). Numerical Treatment of Partial Differential Equations. Springer Science & Business Media. p. 23. ISBN 978-3-540-71584-9.

(Schneider, 2020) ⇒ John B. Schneider (2020). " Understanding the Finite-Difference Time-Domain Method". In: School of electrical engineering and computer science Washington State University.
- QUOTE: The FDTD method makes approximations that force the solutions to be approximate, i.e., the method is inherently approximate. The results obtained from the FDTD method would be approximate even if we used computers that offered infinite numeric precision.

(Causon & Mingham, 2010) ⇒ D. M. Causon, and C. G. Mingham (2010). "Introductory Finite Difference Methods for PDEs". In: Bookboon.
- QUOTE: This book provides an introduction to the finite difference method (FDM) for solving partial differential equations (PDES). In addition to specific FDM details, general concepts such as stability, boundary conditions, verification, validation and grid independence are presented which are important for anyone wishing to solve PDEs by using other numerical methods and/or commercial software packages. Materials presented in order of increasing complexity and supplementary theory is included in appendices.