Discontinuous Galerkin approximations of the Jordan-Moore-Gibson-Thompson equation in the vanishing relaxation limit

The Jordan-Moore-Gibson-Thompson (JMGT) equation models nonlinear acoustic wave propagation in thermally relaxing media and in the vanishing relaxation limit approaches the damped Westervelt equation. We investigate discontinuous Galerkin spatial discretizations of the JMGT equation on simplicial meshes and analyze their behavior uniformly with respect to the relaxation parameter. Under practically relevant mixed Neumann and absorbing boundary conditions, we derive a priori error estimates independent of the relaxation parameter. These estimates enable a rigorous singular limit analysis, yielding convergence of the semi-discrete JMGT approximations to the corresponding Westervelt pressure profile at a linear rate. This also sheds light on the expected behavior of exact solutions in the vanishing relaxation limit. For the fully discrete problem, we propose a Newmark-type method based on a reformulation as a coupled second-/first-order system. Numerical experiments support the theoretical findings and demonstrate the robustness of the approach in the small-parameter regime.

Source: arXiv:2606.14655v1 - http://arxiv.org/abs/2606.14655v1 PDF: https://arxiv.org/pdf/2606.14655v1 Original Link: http://arxiv.org/abs/2606.14655v1