Limiter-Free Third Order Logarithmic Reconstruction

Abstract

A third order conservative reconstruction, in the context of finite volume schemes for hyperbolic conservation laws, is constructed based on logarithmic functions. This logarithmic method reconstructs without the use of a limiter, any preprocessing of input data, special treatments for local extrema, or shock solutions. Also the method is local in the sense that data from only the nearest neighbors are required.We test the new reconstruction method in several numerical experiments, including nonlinear systems in one and two space dimensions.

Compact third-order limiter functions for finite volume methods

“…In contrast to the method based upon a polynomial approximation of the interface values, Artebrant and Schroll [5] have recently developed a new non-polynomial reconstruction function. Originally motivated by the work of Marquina [18], who used a hyperbola as reconstruction function, they recover the interface values with a local double logarithmic (LDLR) Ansatz function.…”

On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas

“…There is also relevant recent work on extensions of hyperbolic reconstructions in[1]. Applications of this new kind of reconstructions within the HMRoe framework are currently under investigation.The main reasons for choosing hyperbolic reconstructions are listed below:…”

A Gas Dynamics Method Based on the Spectral Deferred Corrections (SDC) Time Integration Technique and the Piecewise Parabolic Method (PPM)

“…We used 400 grid points and Figure 7 shows our computational result. We obtained highly resolved solution comparable to [8] and [12,14]. This is an important problem to study, since the solution has some structure in the smooth region.…”

“…There exists number of other high-resolution schemes such as ENO [8], WENO [9], TVD [10], FCT [11], PHM [12], and LLR [13,14] methods. In several instances, comprehensive studies have been carried out to compare the performance of these schemes [15][16][17][18][19].…”