Compact third-order limiter functions for finite volume methods

We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to non-linear 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 [R. Artebrant, H.J. Schroll, Limiter-free third order logarithmic reconstruction, SIAM J. Sci. Comput. 28 (2006) 359-381].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 limiter function. The limiter belongs to a family of functions, which are based upon a non-polynomial and non-linear reconstruction 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≲1.6)(CFL≲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-hydrodynamic equations, confirm a significant improvement of shock resolution, high accuracy for smooth functions and computational efficiency.