Weak periodic solutions and numerical case studies of the Fornberg-Whitham equation

Abstract

Spatially periodic solutions of the Fornberg-Whitham equation are studied to illustrate the mechanism of wave breaking and the formation of shocks for a large class of initial data. We show that these solutions can be considered to be weak solutions satisfying the entropy condition. By numerical experiments, we show that the breaking waves become shock-wave type in the time evolution.

Solution concepts, well-posedness, and wave breaking for the Fornberg–Whitham equation

“…Well-posedness and L 1 -stability for periodic weak entropy solutions to the Fornberg-Whitham equation has been shown independently in [22,Sect. 2] along the lines of Kružkov's original paper [24] and with an adaptation of an older technique by Fujita and Kato for the Navier-Stokes equation based on the analytic semigroup generated by −ε∂ 2…”

“…Some numerical case studies of wave breaking as the formation of shocks in weak solutions on the torus are contained in [22]. They suggest that only negative infinities of u x are developing and that u x stays bounded from above at the moment of wave breaking.…”

“…is in contradiction to (21), we may then conclude that u(x, t) = v(x −ct) cannot define a weak solution of the Fornberg-Whitham equation.To prove (24), we first note that due to(22) the distribution (1− d 2 dξ 2 )((v−c) 2 /2) +v has support in the singleton set {0}, thus equals a finite linear combination of derivatives of the Dirac distribution δ (concentrated at ξ = 0). Recall that v is globally continuous, even C 3 outside ξ = 0, and by (23) the derivative v is locally integrable; hence also((v − c) 2 /2) = (v − c)vis locally integrable.…”

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

“…However, there is no exact solution for these two initial datums. Compared with the results in [13], our dissipative scheme D1 and Using the initial condition of the KdV equation in [2] as…”

“…There are not many numerical schemes for the Fornberg-Whitham type equation. In [13], the finite difference method is adopted to solve the shock solution. The authors did some valuable numerical analysis by the discontinuous Galerkin method in [16], in which the comparisons has been made between the conservative scheme and dissipative scheme, as well as theoretical analysis.…”