Several generalizations of the relativistic models of Burgers equations have recently been established and developed on different spacetime geometries. In this work, we take into account the de Sitter spacetime geometry, introduce our relativistic model by a technique based on the vanishing pressure Euler equations of relativistic compressible fluids on a (1+1)-dimensional background and construct a second order Godunov type finite volume scheme to examine numerical experiments within an analysis of the cosmological constant. Numerical results demonstrate the efficiency of the method for solutions containing shock and rarefaction waves.
The relativistic versions of Burgers equations on the Schwarzschild, FLRW, and de Sitter backgrounds have recently been derived and analyzed numerically via finite volume approximation based on the concerned models. In this work, we derive the relativistic Burgers equation on a Schwarzschild-(anti-)de Sitter spacetime and introduce a secondorder Godunov-type finite volume scheme for the approximation of discontinuous solutions to the model of interest. The effect of the cosmological constant is also taken into account both theoretically and numerically. The efficiency of the method for solutions containing shock and rarefaction waves are presented by several numerical experiments.
A relativistic generalization of the inviscid Burgers equation was introduced by LeFloch and co-authors and was recently investigated numerically on a Schwarzschild background. We extend this analysis to a Friedmann-Lemaître-Robertson-Walker (FLRW) background, which is more challenging due to the existence of time-dependent, spatially homogeneous solutions. We present a derivation of the model of interest and we study its basic properties, including the class of spatially homogeneous solutions. Then, we design a second-order accurate scheme based on the finite volume methodology, which provides us with a tool for investigating the properties of solutions. Computational experiments demonstrate the efficiency of the proposed scheme for numerically capturing weak solutions.
Abstract. This study describes the mathematical construction of a real-life model by means of parametric equations, as well as the two-and three-dimensional visualization of the model using the software GeoGebra. The model was initially considered as "determining the parametric equation of the curve formed on a plane by the point of a pen, positioned on an obstacle of height h, during the process of raising the pen vertically to the surface by linearly moving its backend on the surface." Firstly a solution was sought for this problem in two dimensions. Based on this problem, two additional sub-problems were formed on a plane, and parametric equations were calculated for these sub-problems as well. The curves formed by these parametric equations were then visualized using GeoGebra. In the second stage, the model was improved, and the parametric equation of the curve formed in the space by the pen point as a result of moving the pen's back-end along any function was determined. The curve formed by this parametric equation was also visualized using the GeoGebra 3-D environment. It is expected that determining mathematical concepts and relationships based on real-life models with these types of training tasks, as well as jointly considering the algebraic and geometric representations during the process, will improve the students' perceptions relating to mathematics.
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