Asymptotic and statistical models have been the only practical means, in terms of cost, performance and accuracy, for simulating electrically large environments. We have developed a Full-Wave Time-Domain simulation on commodity Graphics Processing Units (GPUs) employing higher-order Scaling-Function based Multi-Resolution Time-Domain (S-MRTD) techniques. Developed almost ten years ago, S-MRTD techniques are exceptionally memory efficient but have been under-utilized because of their extra computational complexity. We show on GPUs, that besides permitting the optimal utilization of their relatively limited memory resources, this technique's classic shortcoming of increased processing per cell is actually advantageous. The result is a simulator of unprecedented low-cost and high-fidelity, that is orders of magnitude faster (134x) than existing ones of comparable accuracy, and applicable to problems not otherwise possible.