A filtered Chebyshev spectral method for conservation laws on network

We propose a spectral method based on the implementation of Chebyshev polynomials to study a model of conservation laws on network. We avoid the Gibbs phenomenon near shock discontinuities by implementing an exponential filter in the frequency space coupled with the Midpoint method for time marching. This strategy consists in adding a local viscosity into the model, and is suitable to contrast the spurious oscillations appearing in the profile of the solution. We prove the convergence of the semi-discrete method by using the compensated compactness theorem and exploiting the regularity of viscous profiles. Thanks to several simulations, we make a comparison between the implementation of the proposed method with a first order finite volume scheme.