Dornhorst

Dornhorst, also sometimes referred to as the Dornhorst stability technique, is a numerical method used in computational fluid dynamics (CFD) and heat transfer for accelerating the convergence of iterative solutions, particularly for coupled or implicitly solved problems. It is often employed when dealing with equations involving nonlinearities or strong coupling between different physical phenomena.

The core idea behind the Dornhorst method is to introduce a relaxation factor, or a series of relaxation factors, that modifies the iterative updates. This allows for larger adjustments to the solution variables while maintaining stability. The method dynamically adjusts these relaxation factors during the iterative process, typically based on the observed convergence behavior of the solution.

Specifically, the Dornhorst method aims to dampen oscillations and prevent divergence in the solution, which can occur in standard iterative schemes when faced with strong nonlinearities or coupling. By carefully tuning the relaxation factors, the method can often significantly reduce the number of iterations required to achieve a converged solution compared to traditional iterative techniques like Gauss-Seidel or Jacobi methods.

The practical implementation of the Dornhorst method involves monitoring the residual errors or solution updates at each iteration and then adjusting the relaxation factors accordingly. The specific strategy for adjusting these factors can vary, and different approaches may be more suitable depending on the characteristics of the problem being solved. Finding the optimal strategy often involves some experimentation and tuning.

While the Dornhorst method can offer significant improvements in convergence speed, it is not a universal solution for all numerical problems. The effectiveness of the method depends on the nature of the equations being solved, the chosen discretization scheme, and the specific parameters used in the relaxation factor adjustment strategy. In some cases, other acceleration techniques, such as multigrid methods or Krylov subspace methods, may prove more effective.