A study on high order numerical method for solving hyperbolic conservation laws : 쌍곡 보존 법칙들을 풀기 위한 고차정확도 수치기법에 대한 연구

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자연과학대학 수리과학부
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서울대학교 대학원
Finite difference WENOWavelet analysisGrid adaptationEuler equationIdeal MHD equationEfficient and simple numerical method
학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 강명주.
In this thesis, we develop efficient and high order accurate numerical schemes for solving hyperbolic conservation laws such as the Euler equation and the ideal MHD(Magnetohydrodynamics) equations. The first scheme we propose is the \textit{wavelet-based adaptive WENO method}. The Finite difference WENO scheme is one of the popular numerical schemes for application to hyperbolic conservation laws. The scheme has high order accuracy, robustness and stable property. On the other hand, the WENO scheme is computationally expensive since it performs characteristic decomposition and computes non-linear weights for WENO interpolations. In order to overcome the drawback, we propose the adaptation technique that applies WENO differentiation for only discontinuous regions and central differentiation without characteristic decomposition for the other regions. Therefore continuous and discontinuous regions should be appropriately classified so that the adaptation method successfully works. In the wavelet-based WENO method, singularities are detected by analyzing wavelet coefficients. Such coefficients are also used to reconstruct the compressed informations.

Secondly, we propose \textit{central-upwind schemes with modified MLP(multi-dimensional limiting process)}. This scheme decreases computational cost by simplifying the scheme itself, while the first method achieve efficiency by skipping grid points. Generally the high-order central difference schemes for conservation laws have no Riemann solvers and characteristic decompositions but tend to smear linear discontinuities.
To overcome the drawback of central-upwind schemes, we use the multi-dimensional limiting process
which utilizes multi-dimensional information for slope limitation to control the oscillations across discontinuities for multi-dimensional applications.
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College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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