Digital twins are transforming the way we monitor, analyze, and control physical systems, but designing architectures that balance real-time responsiveness with heavy computational demands remains a challenge. Cloud-based solutions often struggle with latency and resource constraints, while edge-based approaches lack the processing power for complex simulations and data-driven optimizations. To address this problem, we propose the High-Precision High-Performance Computer-enabled Digital Twin (HP2C-DT) reference architecture, which integrates High-Performance Computing (HPC) into the computing continuum. Unlike traditional setups that use HPC only for offline simulations, HP2C-DT makes it an active part of digital twin workflows, dynamically assigning tasks to edge, cloud, or HPC resources based on urgency and computational needs. Furthermore, to bridge the gap between theory and practice, we introduce the HP2C-DT framework, a working implementation that uses COMPSs for seamless workload distribution across diverse infrastructures. We test it in a power grid use case, showing how it reduces communication bandwidth by an order of magnitude through edge-side data aggregation, improves response times by up to 2x via dynamic offloading, and maintains near-ideal strong scaling for compute-intensive workflows across a practical range of resources. These results demonstrate how an HPC-driven approach can push digital twins beyond their current limitations, making them smarter, faster, and more capable of handling real-world complexity.