Application of a new class of high accuracy TVD schemes to the Navier-Stokes equations

An analysis of finite-difference and finite-volume formulations of conservation laws

“…Using the fact that P 1 + P 2 + P 3 = I = (36) where I is the unit matrix, one can express the other two projection matrices as…”

Positivity preserving pointwise implicit schemes with application to turbulent compressible flat plate flow

“…Therefore, what is commonly done in steady state calculations is to apply the one-dimensional TVD schemes to each of the co-ordinate directions separately (operator splitting), which is also the approach adopted in the present work. There is, however, no mathematical theory to guarantee that this approach leads to oscillation-free solutions, although it has been observed to be effective in inviscid flow computations [8].…”

“…With regard to accuracy, Reference [9] employs a second-order accurate TVD scheme and Reference [11] a third-order accurate one. In Reference [10] it is the Chakravarthy -Osher family of TVD schemes (also given in References [4,8,18,19]) that is used and it will be employed in the present research too in order to perform turbulent flow computations because: (i) a family of high-order accurate TVD schemes can be generated by merely varying a single parameter and (ii) it also includes a third-order accurate TVD scheme, which would be the ideal candidate to use. In general, advection schemes that are accurate to odd orders are dissipative and dissipative truncation errors help towards damping-out oscillations.…”

On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations

“…Since, such a TVD scheme in multidimensions are at most first order accurate [9], the extension of TVD schemes to multi-dimensions are usually carried out in a dimension-by-dimension manner [6]. However, such dimensional splitting is not possible for unstructured grids.…”