Mathematical and Numerical Aspects of Scalar Nonlinear Conservation Laws
Master thesis
Permanent lenke
https://hdl.handle.net/11250/3089868Utgivelsesdato
2023Metadata
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- Studentoppgaver (TN-IMF) [104]
Sammendrag
This thesis delves deeply into numerical solutions to nonlinear conservationlaws. It focuses largely on the Method of Characteristics, its application insolving conservation laws, and the implementation of solutions in MATLABusing the Lax-Friedrichs scheme.The first chapter offers numerous examples of linear conservation rules andexamines its numerical scheme. The equations’ stability qualities are care-fully examined. In this chapter, the Method of Characteristics is extensivelyused to solve conservation laws, and the efficiency of the Upwind Scheme isproved.The generic solution to the nonlinear conservation law, u_t + f (u)_x = 0, isinvestigated in Chapter 2. The Characteristics Method is used to deduce thecriteria for u_t + f (u)_x = 0 and to examine the Lax-Friedrichs scheme. TheRankine-Hugoniot condition, the development of similarity and shock wavesolutions, and the distinctions between convex and concave flux are all cov-ered in this chapter. It also provides a thorough comparison of the Method ofCharacteristics and the Finite Difference Technique. The chapter concludeswith an examination of inadequate conservation legislation remedies.The third chapter tackles a more complicated Riemann issue, presenting athorough solution as well as MATLAB-based visualization. The use of priorchapters’ knowledge and methodologies to this more complicated issue illus-trates the methods’ adaptability and robustness.In conclusion, this thesis makes an important addition to the understand-ing and application of the Method of Characteristics and the Lax-Friedrichsscheme in the solution of nonlinear conservation laws, as evidenced by prac-tical MATLAB implementation. The comparison of numerical systems, aswell as the expansion to complicated Riemann problems, improve the study’svalue in improving numerical analysis in general