We present a new method for near-source ground-motion calculations due to earthquake rupture on potentially geometrically complex faults. Following the recently introduced Discontinuous Galerkin approach with local time stepping on tetrahedral meshes, we use piecewise polynomial approximations of the unknown variables inside each element and achieve the same approximation order in time and space due to the new ader time integration scheme that uses Arbitrary high-order DERivatives. We show how an external source term and its heterogeneous properties in space and time, given by a fine discretization of an extended rupture surface, can be included in much coarser tetrahedral meshes due to the subcell resolution of the high-order polynomial representation. Hereby, the rupture surface is represented very generally as a point cloud of the center locations of individual subfaults at which each polynomial test function is evaluated exactly inside an element and the space– time integration of the source term is accurately computed at each timestep. Besides the incorporation of complex source kinematics we also present the effects of model boundaries that can degrade the accuracy of seismograms due to weak artificial reflections. We propose an extended computational domain of a coarsely meshed buffer region and show that our scheme using the local timestepping completely avoids such boundary problems with only slightly increasing the computational cost. We validate the new approach against different test cases, comparing our results with analytic, quasianalytic, and a series of reference solutions. Our work shows that adding the functionality of accurately treating finite source-rupture models into the general framework of the ader-Discontinuous Galerkin approach is an important contribution to modeling realistic earthquake scenarios, allowing the efficient inclusion of heterogeneous source kinematics and complex rupture-surface geometries in near-source ground-motion simulations.