Non-linear fourth-order image interpolation for subpixel edge detection and localization
Image and Vision Computing
Non-oscillatory central-upwind scheme for hyperbolic conservation laws
International Journal of Computational Fluid Dynamics
On Stability, Monotonicity, and Construction of Difference Schemes II: Applications
SIAM Journal on Scientific Computing
Applied Numerical Mathematics
Advances in Engineering Software
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The aim of this work is to solve hyperbolic conservation laws by means of a finite volume method for both spatial and time discretization. We extend the ideas developed in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779; X.-D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425] to fourth-order upwind and central schemes. In order to do this, once we know the cell-averages of the solution, $\overline {u}_j ^n$, in cells $I_{j}$ at time $T=t^n$, we define a new three-degree reconstruction polynomial that in each cell, $I_{j}$, presents the same shape as the cell-averages $\{ {\overline {u}_{j-1} ^n,\overline {u}_j ^n,\overline {u}_{j+1} ^n}\}$. By combining this reconstruction with the nonoscillatory property and the maximum principle requirement described in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779] we obtain a fourth-order scheme that satisfies the total variation bounded (TVB) property. Extension to systems is carried out by componentwise application of the scalar framework. Numerical experiments confirm the order of the schemes presented in this paper and their nonoscillatory behavior in different test problems.