CIP methods for hyperbolic system with variable and discontinuous coefficient

TL;DR

This paper introduces a multi-moment scheme for 1D hyperbolic equations with smooth and discontinuous coefficients, utilizing backward characteristics, Hermite interpolation, and immersed interface methods for efficient and accurate solutions.

Contribution

It presents a novel multi-moment method combining backward characteristic approach with cubic Hermite interpolation and immersed interface techniques for hyperbolic systems with variable and discontinuous coefficients.

Findings

Exact update formulas for solution and derivatives.

Effective handling of discontinuities in wave speed.

Extension to Maxwell's equations with variable materials.

Abstract

We propose a multi-moment method for one-dimensional hyperbolic equations with smooth coefficient and piecewise constant coefficient. The method is entirely based on the backward characteristic method and uses the solution and its derivative as unknowns and cubic Hermite interpolation for each computational cell. The exact update formula for solution and its derivative is derived and used for an efficient time integration. At points of discontinuity of wave speed we define a piecewise cubic Hermite interpolation based on immersed interface method. The method is extended to the one-dimensional Maxwell's equations with variable material properties.