A Genuinely High Order Total Variation Diminishing Scheme for One-Dimensional Scalar Conservation Laws

2010 ◽  
Vol 48 (2) ◽  
pp. 772-795 ◽  
Author(s):  
Xiangxiong Zhang ◽  
Chi-Wang Shu
Keyword(s):  
Conservation Laws ◽  
Total Variation ◽  
High Order ◽  
Scalar Conservation Laws ◽  
Total Variation Diminishing ◽  
One Dimensional ◽  
Variation Diminishing ◽  
Scalar Conservation

One-Dimensional Scalar Conservation Laws with Regulated Data

2020 ◽  
Vol 52 (3) ◽  
pp. 3114-3130
Author(s):  
Helge Kristian Jenssen ◽  
Johanna Ridder
Keyword(s):  
Conservation Laws ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Scalar Conservation

scholarly journals Fractional BV spaces and applications to scalar conservation laws

2014 ◽  
Vol 11 (04) ◽  
pp. 655-677 ◽  
Author(s):  
C. Bourdarias ◽  
M. Gisclon ◽  
S. Junca
Keyword(s):  
Conservation Laws ◽  
Stability Result ◽  
Entropy Solutions ◽  
Smoothing Effect ◽  
Scalar Conservation Laws ◽  
One Dimensional ◽  
Regular Functions ◽  
Bv Functions ◽  
Scalar Conservation ◽  
First Time

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BVs(ℝ) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVs spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVs initial data. Furthermore, for the first time, we get the maximal Ws,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

scholarly journals Development of High-Resolution Total Variation Diminishing Scheme for Linear Hyperbolic Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Rabie A. Abu Saleem ◽  
Tomasz Kozlowski
Keyword(s):  
High Resolution ◽  
Total Variation ◽  
Initial Conditions ◽  
Hyperbolic Problems ◽  
Total Variation Diminishing ◽  
One Dimensional ◽  
Flux Limiter ◽  
Variation Diminishing ◽  
Flux Limiters ◽  
Theta Method

A high-resolution, total variation diminishing (TVD) stable scheme is derived for scalar hyperbolic problems using the method of flux limiters. The scheme was constructed by combining the 1st-order upwind scheme and the 3rd-order quadratic upstream interpolation scheme (QUICK) using new flux limiter function. The new flux limiter function was established by imposing several conditions to ensure the TVD properties of the scheme. For temporal discretization, the theta method was used, and values for the parameter θ were chosen such that the scheme is unconditionally stable. Numerical results are presented for one-dimensional pure advection problems with smooth and discontinuous initial conditions and are compared to those of other known numerical schemes. The results show that the proposed numerical method is stable and of higher order than other common schemes.

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