TVD Scheme for SUIKOU-2D v2.0

TVD stands for Total Variation Diminishing, a scheme to eliminate the numerical oscillations or commonly occurred in the explicit numerical methods when they are approaching the high gradient regions. The mathematical expressions of TVD can be reviewed here. I am not going to explain detail about TVD but systematically outlining its implementation to the current version of SUIKOU-2D.
As in previous post, SUIKOU-2D v1.0 was developed to solve full set of Shallow Water Equations (SWEs) using the explicit MacCormack Method . Artificial viscosity (AV) scheme, a control of numerical dissipations to handle shock waves was added. AV is acting as a fashioned approach where the viscosity/turbulence parameters are given as fixed values to all computational steps. It takes series of iterations to get a convergence result. Several improvement efforts has been sought and now it's being altered by TVD scheme in the latest codes of SUIKO-2D. The convergence is quite faster compared to previous one. The general computation blocks are drawn as below figure.

In SUIKOU v2.0 a five-points symmetric TVD term has been used to remove the numerical oscillations and calculated after the corrector step.

where r( ) is scalar product of two vector Ui+1/2 and Ui-1/2 within the point brackets below:

in which U=(h,u,v)

these scalar products can be rewritten in detail as

The function G( ) is defined as

while the flux limiter is:

and C variable is a function of courant number, Cr

The imaginary grid points of above variables are described as below:

Back to AV, it's obviously seen that it is likely a simple model of TVD. The corrections are only made on the basis of water depth, h, whilst TVD deploys all flow variables (u,v,h) to furnish final results.