the development of convection term discretisation

1. According 2-order accurate FVM, the consistent convection discretisation scheme - 2-order central scheme is a suitable choice.
   However, when combined with explicit temporal discretisation, this scheme is unconditionally unstable.
2. 1-order accurate scheme, such as Upwind Scheme, is adopted in order to achieve stability.
   However, this scheme overcomes aformentioned problem by introduing massive diffusion.Namely, it reduces the accuracy.

• Summary: 1 and 2 is about how to achieve stability and accuracy simultaneously.

Lax-Wendroff family schemes are proposed. 2-order accuracy and stability are achieved by combing spatial and temporal discretisation.
However, in the case of steay-state calculation, these schemes introduce an unrealistic dependence of the solution on the time-step.

• Summary: They can achieve stability and accuray at the same time, but they cause another problem ( I don’t understand).

A family of 2-order schemes with independent time integration are developed by Beam and Warming.
However, these schemes cause unphysical oscillations in the solution which severely reduce its quality. Outside of physically meaningful boundedness( unboundedness) is unacceptable.

• Summary: They achieves stability and accuracy all at once and they overcome the problem in 3, but they cause another problem - unboundedness.

1. 4-order artificial dissipation is introduced to achieve boundedness.
   However, it reduces the accuracy and it can’t guarantee boundedness.

2. A series of possible solutions are proposed to achive boundedness and accuracy. Famous schemes are upwind-biased schemes, switching schemes and blending schemes.
   However, they can only conditionly guarantee accuray and boundedness.

• Summary: 5 and 6 can’t guarantee accuracy and boundedness simultaneously and completely.

Flux-limiting schemes arise. They are classied as “shock-capturing schemes”, eventually relulting in TVD schemes. These schemes struct a flux-limiter which is dependent on local flow properties, thus introducing a non-linear dependence of the solution.One of the main conclusions of the TVD analysis is that a scheme has to be non-linear in order to be bounded and more than 1-order accurate.
Howerver, though these schemes behave well at sharp profiles, they are still too diffusive for smooth profiles. And they may cause problem-convergence.

• Summary: TVD schems offer reasonably good accuracy and at the same time guarantee boundedness, but accuray can be improved for smooth profiles.

In order to develop a scheme which can behave well both for sharp and smooth profiles, the Normalised Variable Approach (NVA) is introduced.
However, it can’t guarantee any convergence.

• Summary: Good accuray and boundedness are achieved at the expense of convergence.

……Now, it’s you turn to contribute.

做了点微小的工作，谢谢大家。