A New Approximation (ziti's δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions: Applications to Convection, Burgers, Gas Dynamics and Some Biological Problems

Larbi BSISS^1, and Cherif ZITI¹

¹Department of Mathematics, University Moulay Ismail, Meknes, Morocco

Cite this paper:
Larbi BSISS and Cherif ZITI. A New Approximation (ziti's δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions: Applications to Convection, Burgers, Gas Dynamics and Some Biological Problems. Turkish Journal of Analysis and Number Theory. 2016; 4(4):98-108. doi: 10.12691/tjant-4-4-3

Abstract

As it is well known in numerical analysis, most of the numerical schemes have undesirable oscillations, especially near the domain's border, or near the physical phenomena (empty region, collapse, boundary layer, among others) (mathematically invisible) eg: Burgers equation(the solution loses its regularity in finite time). In the case where the differential problem solution presents a singularity (shock, blow-up which cannot be numerically detected easily), the classical scheme cannot generally operate correctly and in the best case we are confronted with a very difficult algorithm, especially in several dimensions. Our objective here is to construct a less complicated scheme compared to the classical methods by keeping their advantages and obtained the admissible solution in the most difficult situations without complications obtained from the selected meshing. In this paper, we applied our new method called ziti's δ- scheme which is able to resist to such oscillations near the singularity and enables us to detect a lot of physical phenomena (eg: shock waves, rarefaction waves, conservation of the matter quantity ...). We depict the ziti's δ- scheme for the multidimensional partial differential equations and systems on any meshing with simple numbering. We apply our method to some models and compare its results with the exact one and other classical numerical methods. We can conclude that our results are very striking. The ziti's δ-method that we obtained is faster and more efficient and robust.

Keywords:
hyperbolic Riemann problem entropy solution shock Gibbs phenomena Godunov Grimm oscillation ziti's δ- scheme

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References:

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