Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

n this paper we present a low-order numerical scheme developed using the Proper Orthogonal (POD) method to address non-homogeneous parabolic equations in both one and two dimensions. The proposed schemes leverage the POD technique to reduce the computational complexity associated with solving these equations while maintaining accuracy. By employing POD, the high-dimensional problem is approximated by a reduced set of models, allowing for a more efficient representation of the system dynamics. The application of this method to non-homogenous parabolic equations offers a promising approach for enhancing the computational efficiency of simulations in diverse fields, such as fluid dynamics, heat conduction, and reaction-diffusion processes. The presented numerical scheme demonstrates its efficacy in achieving accurate results with significantly reduced computational costs, making it a valuable tool for applications demanding efficient solutions to non-homogeneous parabolic equations in one and two dimensions.