Shock-Fitting Vs. Shock-Capturing: Why Temperature Results Differ?

When simulating high-speed compressible flows, computational fluid dynamics (CFD) offers two primary approaches for handling shock waves: shock-fitting and shock-capturing. While both methods aim to accurately model these phenomena, they employ fundamentally different techniques, which can sometimes lead to varying results. This article delves into the core differences between shock-fitting and shock-capturing methods, explores why discrepancies in results may arise, and provides insights into interpreting these differences within the context of CFD simulations.

What are Shock-Fitting and Shock-Capturing?

To fully grasp the potential for result variation, it’s essential to understand the unique approach of each method.

Shock-Fitting: Precision at the Cost of Complexity

Shock-fitting is a numerical technique where shock waves are treated as sharp discontinuities and explicitly tracked as they propagate through the computational domain. This method requires precise knowledge of the shock's location at each time step. The computational grid is typically conformed to the shock, meaning the grid points are aligned with the shock front. This allows for highly accurate representation of the shock itself, minimizing numerical diffusion and preserving the sharp gradients across the discontinuity.

The key advantage of shock-fitting lies in its ability to resolve shock waves with exceptional accuracy. By treating the shock as a boundary, the method avoids smearing the discontinuity over multiple grid cells, which can occur in other methods. This makes it particularly suitable for problems where the precise location and strength of the shock are critical, such as in the analysis of supersonic flows or detonations.

However, shock-fitting also comes with significant drawbacks. It is considerably more complex to implement than shock-capturing, as it requires sophisticated algorithms for tracking and adapting the grid to the moving shock. Furthermore, shock-fitting struggles with complex flow scenarios involving multiple interacting shocks or shocks interacting with other flow features, such as vortices or turbulent structures. Each interaction requires careful handling and can significantly increase the computational cost. Therefore, while shock-fitting excels in specific scenarios, its complexity and limitations make it less versatile for general-purpose CFD simulations.

Shock-Capturing: Robustness and Versatility

In contrast to shock-fitting, shock-capturing methods do not explicitly track shock waves. Instead, they rely on the numerical scheme itself to capture the shock as a steep gradient within the solution. This is achieved by using special numerical techniques, such as flux-limiters or high-resolution schemes, which are designed to minimize oscillations and maintain stability in the presence of discontinuities.

Shock-capturing methods solve the governing equations (e.g., Euler or Navier-Stokes equations) across the entire computational domain, including the shock region. The shock wave is not treated as a boundary but rather as a region of rapid change in flow properties, such as pressure, density, and temperature. The numerical scheme then approximates this rapid change over a few grid cells, effectively