The nonlinear flux limiter approach employed in the proposed discretization scheme has several potential limitations that could affect its robustness and accuracy:
Nonlinearity and Differentiability: Many nonlinear flux limiters are not differentiable, which poses challenges for the Jacobian computation in Newton's method. This can lead to difficulties in convergence for the nonlinear solver, particularly in regions where the temperature gradients are steep or where the flux limiter is highly active.
Sensitivity to Parameter Choices: The performance of nonlinear flux limiters can be sensitive to the choice of parameters, such as the threshold values that determine when the limiter is activated. Poorly chosen parameters can lead to excessive numerical diffusion or oscillations in the solution, undermining the accuracy of the simulation.
Limited Applicability to Complex Geometries: While the current implementation of the SMART scheme is effective in many scenarios, it may struggle in highly complex geometries or under extreme anisotropic conditions. The assumptions made in the formulation of the limiter may not hold, leading to inaccuracies.
To enhance the robustness and accuracy of the nonlinear flux limiter approach, several improvements can be considered:
Development of Adaptive Limiters: Implementing adaptive flux limiters that can adjust their behavior based on local flow conditions and gradients could improve performance. These limiters could switch between different formulations depending on the local characteristics of the solution, providing better control over numerical diffusion and oscillations.
Hybrid Approaches: Combining nonlinear limiters with other stabilization techniques, such as artificial viscosity or shock-capturing methods, could help mitigate the limitations of the flux limiter approach. This hybridization could provide a more comprehensive solution strategy that balances accuracy and stability.
Robustness Testing: Conducting extensive robustness testing across a range of scenarios, including varying anisotropies and complex geometries, can help identify weaknesses in the current limiter approach. This testing can inform the development of more resilient limiter formulations that maintain accuracy under diverse conditions.
By addressing these limitations and implementing these improvements, the nonlinear flux limiter approach can be made more effective, enhancing the overall performance of the discretization scheme in simulating anisotropic heat transport.