Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation
This work provides a rigorous analysis and filtering strategy for spurious modes in DG discretizations of the wave equation, which is relevant for numerical approximation of control problems.
The authors perform a complete Fourier analysis of the P1 discontinuous Galerkin semi-discretization of the 1-D wave equation, identifying physical and spurious components with vanishing group velocities. They construct wave packets with arbitrarily small propagation velocity and develop filtering mechanisms to restore uniform propagation, with applications to control problems.
We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.