Numerical methods for the quasilinear wave equations: antiplane shearing of nonlinearly elastic bodies

Dr. Dawn Lott-Crampler

Department of Mathematics, University of Maryland

We formulate an efficient numerical algorithm, based on finite-difference approximations and inspired by algorithms from gas dynamics, to treat the quasilinear wave equation w_{tt} = [\alpha(w_x^2+w_y^2) w_x]_x + [\alpha(w_x^2+w_y^2) w_y]_y governing antiplane motions of nonlinearly elastic bodies in two-dimensional domains. We develop robust and effective numerical methods to capture the shocks that arise and we incorporate body-fitted meshes to handle computation in domains with irregular geometries. In the process of validating our procedures we solve the axisymmetric version of this equation in polar coordinates and develop methods for handling its polar singularity.