The discontinuous Galerkin method for fractal conservation laws
This work provides a numerical framework for solving fractal conservation laws, which are relevant for modeling anomalous diffusion and other nonlocal phenomena, though the results are incremental as they extend existing DG methods to a new class of equations.
The authors develop and analyze a discontinuous Galerkin method for fractal conservation laws, proving stability and error estimates for linear equations and a convergence rate for nonlinear cases with piecewise constant elements, supported by numerical results.
We propose, analyze, and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and whenever piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractal conservation laws.