The integral equation method and the Nyström method, while powerful for solving scattering problems, have certain limitations when dealing with rough boundaries or significant contrasts in physical properties between the media.
Rough Boundaries: The accuracy of the Nyström method can be affected by the roughness of the boundary. Highly irregular surfaces may lead to numerical instabilities or convergence issues. In such cases, special treatment or regularization techniques may be required to handle the singularities that arise near rough boundaries.
Contrast in Physical Properties: Large disparities in physical properties, such as vastly different wave speeds or impedance values between the media, can pose challenges for the Nyström method. These contrasts can lead to oscillations in the numerical solutions or slow convergence rates. Adjustments in the discretization scheme or preconditioning methods may be necessary to address these issues.
Limitations of Integral Equations: Integral equations inherently suffer from the curse of dimensionality, especially in higher dimensions or complex geometries. As the problem size increases, the computational cost and memory requirements can become prohibitive. This can restrict the applicability of the method to large-scale scattering problems.
While the Nyström method is effective for many scattering scenarios, understanding these limitations is crucial for optimizing its performance and reliability in challenging situations.