Late-time asymptotic behavior of solutions to hyperbolic conservation laws on the sphere

We consider nonlinear hyperbolic conservation laws posed on a curved geometry, referred to as "geometric Burgers equations" after Ben-Artzi and LeFloch (2007), when the underlying geometry is the sphere and the flux vector field is determined from a potential function. Despite its apparent simplicity, this model exhibits complex wave phenomena which are not observed in absence of geometrical effects. To study the late-time asymptotic behavior of the solutions of this model, we consider a finite volume method based on a generalized Riemann solver. We provide a numerical validation of the accuracy and efficiency of the method in presence of nonlinear waves and a curved geometry and, especially, demonstrate the contraction, time-variation monotonicity, and entropy monotonicity properties. The late-time asymptotic behavior of the solutions is studied and discussed in terms of the properties of the flux. A new classification of the flux vector field is introduced where we distinguish between foliated flux and generic flux, and the character of linearity of the flux which are expected to be sufficient to predict the late-time asymptotic behavior of the solutions. When the flux is foliated and linear, the solutions are transported in time within the level sets of the potential. If the flux is foliated and is genuinely nonlinear, the solutions converge to their (constant) average within each level set. For generic flux, the solutions evolve with large variations which depend on the geometry and converge to constant values within certain "independent" domains on the sphere. The number of constant values depends on curves that "split" the sphere into independent domains. For fluxes which are linear, foliated or generic only on parts of the sphere, combinations of the late-time asymptotic behavior of the solutions can be obtained which depends also on the interaction between the fluxes at boundaries of these parts of the sphere. (C) 2019 Elsevier B.V. All rights reserved.