A non-iterative numerical approach to the solution of the Darcy problem

Abstract

A non-iterative Forecast-Correction (FC) approach based on a sequential combination of an explicit prediction step and an implicit correction step is presented for the numerical solution of the Darcy problem. In the prediction stage, a high-order RKDG/WENO scheme was employed to accurately approximate convective transport and to reliably capture sharp solution gradients. In the correction stage, a single implicit diffusion step was performed using either BDF1 or FIRK with fixed ("frozen") coefficients. Such construction resolved the tension between the requirements of two classes of numerical schemes. On the one hand, explicit high-order convective methods provide high accuracy; however, they are strongly constrained by the Courant-Friedrichs- Lewy condition. On the other hand, implicit diffusion schemes allow for large time steps. However, such schemes typically require iterative procedures increasing computational cost. The proposed approach eliminates per-time-step iterations, thereby achieving constant and predictable computational workload. Numerical experiments demonstrate that the FC scheme remains stable, suppresses spurious oscillations, and delivers acceptable accuracy at reasonable computational expense. The results confirm the practical effectiveness of the FC method as a numerical tool that balances approximation accuracy, computational stability, and efficiency for solving the Darcy problem. The latter is clearly confirmed by the graphs obtained during the numerical modeling of the Darcy problem, demonstrating the advantages of the proposed approach. The conducted research suggests the possibility of efficient calculations on irregular (including unstructured) grids while maintaining the required accuracy.