Grid Oscillations in Finite-Difference Scheme and a Method for Their Approximate Analysis

Abstract

The study focuses on the phenomenon of short-wave (sawtooth) oscillations manifested in some discrete approximations of hyperbolic systems of equations. A technique for the analysis of oscillations is proposed, decomposing the solution into a “smooth” and a “sawtooth” components, followed by application of the known differential approximation method. The new method makes it possible to assess the properties of initial–boundary-value problems and spectral finite-difference problems. The central-difference scheme for the transport equation is investigated in detail, using various boundary conditions that can be optimized. Possible generalizations of the approach to multidimensional and nonlinear problems are suggested.