Dynamics in spectral solutions of Burgers equation

摘要

For low values of the viscosity coefficient, Burgers equation can develop sharp discontinuities, which are difficult to simulate in a computer. Oscillations can occur by discretization through spectral collocation methods, due to Gibbs phenomena. Under a dynamic point of view, these instabilities are related to bifurcations arising to the discretized equation. For different values of discretized points, herein a study is performed of the dynamics and bifurcations occurring in the spectral solutions of Burgers equation with symmetry. Several phenomena are observed, from limit cycles, strange attractors to the presence of bistability with two periodic attractors, with a periodic attractor and a strange attractor and with two strange attractors. Also, other stable equilibrium points can occur, diverse from the ones corresponding to the solution of Burgers equation.