An easy-to-implement parallel algorithm to simulate complex instabilities in three-dimensional (fractional) hyperbolic systems

Abstract

It is well known that the simulation of fractional systems is a difficult task from all points of view. In particular, the computer implementation of numerical algorithms to simulate fractional systems of partial differential equations in three dimensions is a hard task which has no been solved satisfactorily. In this work, we propose a numerical method to solve systems of hyperbolic (fractional o non-fractional) partial differential equations that generalize various known models from physics, chemistry and biology. The scheme is an explicit technique which has the advantage of being easy to implement for any scientist with minimal knowledge on scientific programming. We propose a computer implementation which exploits the advantages of the efficient matrix algebra already available in Fortran and other languages. The algorithm is presented mathematically as well as in pseudo-code, and a raw Fortran implementation of the computer algorithm is provided in the appendix. This code is susceptible to be compiled in parallel using OpenMP, whence it follows that the computer time can be substantially reduced. As application, we provide some illustrative simulations on the formation of Turing patterns in a three-dimensional system of inhibitor–activator substances in physics. The graphs were obtained using functions of Matlab with the numerical outputs generated by our Fortran code.