This paper presents an advanced numerical scheme designed to tackle the complexities of coupled systems of partial differential equations (PDEs), a common challenge in various scientific and engineering disciplines. Traditional methods often struggle with strong interdependencies and non-linear behaviors, leading to computational inefficiencies or inaccuracies. The proposed approach leverages a novel discretization technique combined with an iterative solver, significantly improving convergence rates and stability.
The core of the methodology involves reformulating the coupled system into a block matrix form, allowing for a more efficient solution strategy. Consider a simplified representation of a coupled system involving two primary variables, u and v, influenced by a source term S and a diffusion coefficient D:
∂u⁄∂t = ∇⋅(Du∇u) + αv + Su
∂v⁄∂t = ∇⋅(Dv∇v) + βu + Sv
Here, α and β represent coupling coefficients, and ∇ is the nabla operator. The scheme employs an adaptive time-stepping algorithm, ensuring accuracy while minimizing computational cost, particularly for transient problems.
Key Advantages of the Proposed Scheme:
- Enhanced Stability: The method exhibits superior stability properties, even for stiff systems.
- Improved Accuracy: Higher-order spatial and temporal discretization leads to more accurate solutions.
- Computational Efficiency: Optimized matrix solvers reduce computational time significantly.
- Broad Applicability: Applicable to a wide range of coupled phenomena, including fluid-structure interaction and thermo-mechanical processes.
Numerical experiments, detailed in this document (arXiv 0028.9430), confirm the scheme's robustness and efficiency across various benchmark problems, outperforming existing methods in terms of both speed and precision. This work provides a valuable tool for researchers and engineers dealing with complex multi-physics simulations.