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The Secret of Shockwave Capture: Why Modern Methods Are More Effective than Traditional Methods?
In computational fluid dynamics, the shock capture method is a technique designed for calculating shock waves in inviscid flows. Computing flows involving shock waves is a challenging task because such flows result in sharp, discontinuous changes in flow variables such as pressure, temperature, density, and velocity at the shock. Traditional methods face many challenges, while modern methods offer superior solutions.
Overview of shock wave capture methods
Shock capture methods usually express the governing equations of inviscid flow (i.e. Euler's equations) in conservation form and calculate shock waves or discontinuities as part of the solution. In this method, no special treatment is performed on the shock waves themselves, in contrast to shock wave fitting methods, which explicitly introduce shock waves in the solution using corresponding shock wave relationships (such as the Rankine–Hugoniot relationship).
Predictions from shock capture methods are often not as sharp as traditional methods and may become blurry between several grid elements. This means that unnatural oscillation phenomena may result in numerical calculations caused by strong seismic waves.
The role of Euler’s equation
The Euler equation is the governing equation of inviscid flow. In order to implement the shock wave capture method, the conservation form of Euler's equation must be used. For flows without external heat transfer and work transfer (isoenergetic flows), the conservation form can be used to solve numerically.
Classical and modern shock wave capture methods
From a historical perspective, seismic capture methods can be divided into classical methods and modern seismic capture methods (also known as high-resolution solutions). Modern shock capture methods are typically biased upwind, in contrast to classical symmetric or centered discretizations. The upwind bias difference scheme discretizes the hyperbolic partial differential equations through differentiation based on the flow direction, while the symmetric or central schemes do not consider the directional information of wave propagation.
In the presence of shock waves, different shock wave capture schemes require a certain degree of numerical dissipation to ensure the stability of the calculation, thereby avoiding the formation of unreasonable numerical oscillations.
The classical seismic capture method will give accurate results in the case of smooth and weak seismic solutions, but when strong seismic waves are present, nonlinear instabilities and oscillations on discontinuities may occur. In contrast, modern shock capture methods often exploit nonlinear numerical dissipation, with feedback mechanisms adjusting the amount of artificial dissipation added based on the characteristics of the solution. Ideally, artificial numerical dissipation would only need to be added near shock waves or other sharp features, while regions of the smooth flow field should remain unchanged.
Concrete examples and prospects
The famous classical shock wave capture methods include MacCormack method, Lax-Wendroff method and Beam-Warming method. Examples of modern shock wave capture schemes include the higher-order total variation reduction (TVD) scheme originally proposed by Harten, the flow correction transmission scheme proposed by Boris and Book, and the monotonic upstream center scheme (MUSCL) based on the Godunov method and introduced by van Leer. ), and various basic non-oscillatory schemes (ENO) proposed by Harten et al.
Another important category of high-resolution solutions belongs to the Riemann approximate solvers, proposed by Roe and Osher. These methods have excellent performance in the accuracy of shock wave capture.
Summary
Overall, modern seismic wave capture methods can provide more stable and accurate results when dealing with strong seismic wave problems, which is based on improvements and adjustments to numerical dissipation. The limitations of elements of classical methods have gradually replaced them with modern methods. In future fluid dynamics research, these methods will face new challenges on how to continue to improve computational performance and solution accuracy. Can we find more perfect shock wave capture technology to deal with increasingly complex flow situations?
