Frequency Accurate Finite Difference Methods
Computing Sciences and Computer Engineering
Deriving numerical approximations independently of the partial differential equation in which they appear leaves the discipline of numerical approximations at the experimental trial and error level. Comparison studies are then required to test which approximations work best for a family of equations. The resulting knowledge is not a quantification, but rather an amalgamation of a body of knowledge. Yet this is the current state-of-the-art in numerical models. Such a body of knowledge may provide intuition, but it cannot provide direct guidance. In contrast, the approach developed here provides direct control of the numerical approximation error. While the numerical approximation error has not been eliminated, it has been reshaped to achieve desired results based directly on the physical properties for the PDE to be simulated. Specifically, we formulate numerical approximations in terms of undefined coefficients. We then map these approximations to Frequencyspace where we solve for these undetermined coefficients in a physically meaningful way. The coefficients are then used directly in their respective numerical approximations. We illustrate the technique first on a spatial derivative, then in the context of an evolution equation, where we introduce a time derivative. After expanding the approximations to higher order derivatives, we illustrate the overall method on Burgers Equation.
Advances in the Applications of Nonstandard Finite Diffference Schemes
(2005). Frequency Accurate Finite Difference Methods. Advances in the Applications of Nonstandard Finite Diffference Schemes, 561-614.
Available at: https://aquila.usm.edu/fac_pubs/21262