Compact high-order numerical schemes for scalar hyperbolic partial differential equations

Abstract

Abstract This work is a comparative study on the numerical behavior of compact high-order schemes for discretizing scalar hyperbolic problems. To this aim, we investigate two possibilities: a high-order Hermitian scheme (HUPS), based upon the scalar variable and its first derivatives in space, and a “classical” high-order upwind scheme (CUPS), based upon the scalar variable and a wide numerical stencil, as a reference scheme. To study the theoretical properties of both schemes we use a spectral analysis based on a Fourier transform of the numerical solution. In a one-dimensional context, this analysis enables us to investigate the impact of the “spurious” components generated by the Hermitian schemes. As demonstrated by the spectral analysis, in some circumstances, this “spurious” component may deeply alter the accuracy of the scheme and even, in some cases, destroy its consistency. Thus, by identifying the salient features of a Hermitian scheme, we try to design efficients Hermitian schemes for multi-dimensional purposes. A two-dimensional Fourier analysis and some scalar numerical tests allow to highlight two high-order versions of a Hermitian scheme (HUPS4,5) of which the numerical properties are compared with a classical fifth-order scheme (CUPS5) based upon a wider stencil.