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Compact third order limiter functions for finite-volume-methods

by M. Cada and M. Torrilhon

(Report number 2008-12)

Abstract
We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to nonlinear smooth functions, we introduce a new class of limiter functions for the spatial reconstruction of hyperbolic equations. We therefore employ and generalize the idea of double-logarithmic reconstruction of Artebrant and Schroll [SIAM J. Sci. Comput. 2006]. The result is a class of efficient third-order schemes with a compact three point stencil. The interface values between two neighboring cells are obtained by a single non-linear limiter function. The new methods handle discontinuities as well as local extrema within the standard semi-discrete TVD-MUSCL framework using only a local three point stencil and an explicit TVD Runge-Kutta time marching scheme. The shape preserving properties of the reconstruction are significantly improved, resulting in sharp, accurate and symmetric shock capturing. Smearing, clipping and squaring effects of classical second-order limiters are completely avoided. Computational efficiency is enhanced due to large allowable Courant numbers (CFL $\lesssim$ 1.6), as indicated by the von Neumann stability analysis. Numerical experiments for a variety of hyperbolic partial differential equations, such as Euler equations and ideal magneto-hydro-dynamic equations, confirm an significant improvement of shock resolution, high accuracy for smooth functions and computational efficiency.

Keywords:

@Techreport{CT08_377,
  author = {M. Cada and M. Torrilhon},
  title = {Compact third order limiter functions for finite-volume-methods},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-12.pdf },
  year = {2008}
}

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