Stability conditions for the numerical solution of convection-dominated problems with skew-symmetric discretizations (Q2903026)

The author presents the stability conditions for the numerical solution of convection-dominated problems with skew-symmetric discretizations. The paper is organized as follows. First the definition and the computation of the von Neumann stability are recalled. The linear transport problem, predicting a stability condition of the type \(\delta t \leq C(\delta x / u)^{2r / (2r - 1)}\) with \(r\) an integer, for several schemes are focused. Then the numerical schemes are constructed for which such a stability condition appears for \(r = 1,\,2,\,3,\,4\) and corresponds to exponents equal to \(2,\,\,\frac{4}{3},\,\,\frac{6}{5}\,\,\)and \(\frac{8}{7}\). Finally, it is shown how this stability criterion extends to nonlinear equations and to multicomponent transport equations (including wave equations).