Fuzzy flux limiter schemes for hyperbolic conservation laws

A classic strategy to obtain high-quality discretisations of hyperbolic partial differential equations (PDEs) is to employ a non-linear mixture of two types of approximations. The building blocks for this are a monotone first-order scheme that deals with discontinuous solution features and a higher-order method for approximating the smooth solution parts. The blending is performed by the so-called flux limiter function. In this paper we introduce a novel approach to flux limiter methods. We show that fuzzy logic (FL) is a useful tool to understand and formulate the limiter functions. After introducing the set-up, we verify that a variety of classic flux limiters can easily be interpreted via FL. Then we show how one can improve limiters making use of the developed FL framework. This is done for initial data and PDEs that describe characteristic settings for hyperbolic problems. Our work shows for the first time in the literature that the two different fields of fuzzy logic and numerical methods for PDEs can be brought together with benefit.