A reduced-order peridynamic differential operator for unsteady convection-diffusion problems

The efficient numerical method for simulating unsteady convection-diffusion problems is a crucial research topic in the field of numerical heat transfer. However, the state in which transport processes dominate and diffusion effects are confined to a relatively small area may cause some difficulties in numerical solutions, such as numerical oscillations and excessive diffusion. Therefore, novel ideas and methods are required. The peridynamic differential operator (PDDO) is an innovative numerical method that offers a non-local form of differential equations. Its most notable advantage lies in its ability to handle sharp gradients without requiring special treatment. Nevertheless, its computational efficiency falls short compared to mesh-based numerical methods. To enhance the computational efficiency of PDDO for solving time-dependent problems, we propose a fast and efficient approach that combines PDDO with proper orthogonal decomposition (POD) technique for solving unsteady convection-diffusion problems. We validate the effectiveness and accuracy of this proposed method through three numerical examples, which demonstrate that it achieves high computational accuracy while significantly reducing computation time when solving long-time calculation problems.