Time or space adaptivity for exterior wave problems with gCQ

The application of Lubich's Convolution Quadrature (CQ) for the time discretization of exterior wave problems formulated as boundary integral equations is nowadays well understood . The CQ method is however restricted by construction to use uniform time steps. More recently this limitation of the CQ method has been overcome and the so-called generalized Convolution Quadrature (gCQ) method has been developed, which allows to use a non uniform sequence of time steps. The gCQ method actually decouples time and space integration, allowing in principle to change the spatial grid from one time step to the next one and also to use a quite general sequence of time points. Once the use of nonuniform time grids is possible, the next step is the development of an adaptive scheme and thus of some mechanism to control the error. We address two possible strategies which are based on the available a priori error estimates for the gCQ. We test the performance of the new methods with a scalar model problem arising from the resolution of the wave problem outside a sphere.