A fractional Burgers equation arising in nonlinear acoustics: theory and numerics

Abstract

The study of a fractional Burgers equation arising in nonlinear acoustics is presented. The motivation comes from an elementary model of shock waves in brass wind instruments, that proves useful in musical acoustics. Such a model results from the coupling of a conservative nonlinear system with a dissipative term; here the dissipation is represented by a fractional derivative in time, for which equivalent diffusive representations can be efficiently used: in a first part, strong solutions, weak solutions and energy balances are examined. In a second part, ad hoc numerical schemes are derived, in order to capture all the physical phenomena at stake in the original model, and to get rid, as far as possible, of the spurious numerical effects which are highly undesirable: to this end, conservative schemes for hyperbolic conservation laws, diffusive realizations for the fractional derivatives and integrals, and splitting of the two are being used.