A review on TVD schemes and a refined flux-limiter for steady-state calculations

Self Cite

SUMMARYA refined r-factor algorithm for implementing total variation diminishing (TVD) schemes on arbitrary unstructured meshes, referred to henceforth as a face-perpendicular far-upwind interpolation scheme for arbitrary meshes (FFISAM), is proposed based on an extensive review of the existing r-factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing r-factor algorithms. The performance of the FFISAM, implemented in 10 classical TVD schemes, is evaluated against four two-dimensional pure-advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the 10 classical TVD schemes.

q_{t} = - {(a q)}_{x}

where q = q ( x , t ) denotes the transported variable and α is the advection velocity.…”

Section: Tvd Schemesmentioning

confidence: 99%

“…Integrating Eq. over a control volume C i = [ x i − Δ x /2 , x i + Δ x /2 ], one can obtain the general semi‐discrete flux‐conservative form:

\frac{d q_{i}}{d t} = \frac{- a ((), q_{i + 1 true/ 2} - q_{i - 1 true/ 2})}{normalΔ x}

where the interface value, q i + 1/2 , can be obtained by using the well‐known k ‐schemes, firstly introduced by Van Leer :For uniform grids,

\begin{matrix} left \\ q_{i + 1 / 2} = q_{i} + \frac{1}{2} ψ ({\overset{true˜}{r}}_{i + 1 true/ 2}) (q_{i} - q_{i - 1}) \\ where ψ ({\overset{true˜}{r}}_{i + 1 true/ 2}) = \frac{1 + k}{2} {\overset{r}{true}}_{i + 1 / 2} + \frac{1 - k}{2}, {\overset{r}{true}}_{i + 1 / 2} = \frac{((), q_{i + 1} - q_{i})}{((), q_{i} - q_{i - 1})} \end{matrix}

For non‐uniform meshes,

\begin{matrix} left \\ q_{i + 1 / 2} = q_{i} + \frac{Δ x_{i}}{2} ψ ({\overset{true˜}{r}}_{i + 1 true/ 2}) {((), \frac{\partial q}{\partial x})}_{i - 1 / 2} \\ where ψ ({\overset{true˜}{r}}_{i + 1 true/ 2}) = \frac{1 + k}{2} \overset{}{true} \end{matrix}

…”

Section: Tvd Schemesmentioning

confidence: 99%

({, \begin{array}{c} center & 0 ⩽ ψ ((), {\overset{r}{true}}_{i + 1 / 2}) ⩽ normalmin ((), 2 {\overset{true˜}{r}}_{i + 1 true/ 2}, 2) when {\overset{true˜}{r}}_{i + 1 true/ 2} > 0 \\ ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = 0 when {\overset{true˜}{r}}_{i + 1 true/ 2} ⩽ 0 \end{array})

…”

Section: Tvd Schemesmentioning

confidence: 99%

“…Some of those are outlined in the succeeding paragraphs for the convenience of discussion (Figure ). Piecewise‐linear SS‐TVD flux limiters :Minmod limiter :

ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = normalmax ([], 0, min ((), {\overset{true˜}{r}}_{i + 1 true/ 2}, 1))

Superbee limiter :

ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = normalmax ([], 0, min ((), 2 {\overset{true˜}{r}}_{i + 1 true/ 2}, 1), min ((), {\overset{true˜}{r}}_{i + 1 true/ 2}, 2))

MUSCL limiter :

ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = normalmax ([], 0, min ((), 2 {\overset{true˜}{r}}_{i + 1 true/ 2}, \frac{{\overset{true˜}{r}}_{i + 1 true/ 2} + 1}{2}, 2))

Koren limiter :

ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = normalmax

…”

Section: Tvd Schemesmentioning

confidence: 99%

“…Superbee limiter :

ψ ((), {\overset{r}{true}}_{i + 1 / 2}) = normalmax ([], 0, min ((), 2 {\overset{true˜}{r}}_{i + 1 true/ 2}, 1), min ((), {\overset{true˜}{r}}_{i + 1 true/ 2}, 2))

…”

Section: Tvd Schemesmentioning

confidence: 99%

See 3 more Smart Citations

A refined r‐factor algorithm for TVD schemes on arbitrary unstructured meshes

Zhang

Jiang

Cheng

et al. 2015

Self Cite

A truncation error analysis of third‐order MUSCL scheme for nonlinear conservation laws

Nishikawa

2020

This article is a rebuttal to the claim found in the literature that the monotonic upstream‐centered scheme for conservation laws (MUSCL) cannot be third‐order accurate for nonlinear conservation laws. We provide a rigorous proof for third‐order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's κ‐scheme recovers a cubic solution exactly with κ=1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third‐order truncation error. Finally, noting that the target spatial operator is a cell‐averaged flux derivative, we prove that the leading truncation error of the MUSCL finite‐volume scheme is third‐order with κ=1/3. The importance of the diffusion scheme is also discussed: third‐order accuracy will be lost when the third‐order MUSLC scheme is used with an incompatible fourth‐order diffusion scheme for convection‐diffusion problems. Third‐order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This article is intended to serve as a reference to clarify confusions about third‐order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying high‐order unstructured‐grid schemes in a subsequent article.

A new framework to construct third‐order weighted essentially nonoscillatory weights using weight limiter functions

Parvin

Dubey

2020

A new simple and generic framework is proposed to construct nonlinear weights for third‐order weighted essentially nonoscillatory scheme (WENO) reconstructions. It is done by imposing necessary conditions on nonlinear weights to get a nonoscillatory WENO scheme. These conditions give further insight into the required structure of nonlinear weights to design third‐order WENO schemes. This new framework for WENO weights is completely different from the existing prevailing approaches. Several nonlinear weights using different functions of a smoothness parameter (termed as weight limiter functions) are proposed and analyzed. These new weights by construction guarantee for third‐order accurate nonoscillatory scheme. Numerical results for various benchmark test problems are given and compared with WENO‐JS3 wnd WENO‐Z3 scheme. Computational results show that WENO schemes using proposed weights achieves third‐order accuracy for smooth solution and resolves discontinuities without spurious oscillations.