• Communications of the Korean Mathematical Society
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• v.13 no.3
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• pp.655-681
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• 1998

Discontinuous solutions or interfaces are common in nature, for examples, shock waves or material interfaces. However, their numerical computation is difficult by the feature of discontinuities. In this paper, we summarize the numerical approaches for discontinuities and interfaces appearing mostly in the system of hyperbolic conservation laws, and explain various numerical methods for them. We explain two numerical approaches to handle discontinuities in the solution: shock capturing and shock tracking, and illustrate their underlying algorithms and mathematical problems. The front tracking method is explained in details and the level set method is outlined briefly. The several applications of front tracking are illustrated, and the research issues in this field are discussed.

• Journal of The Korean Astronomical Society
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• v.34 no.4
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• pp.281-283
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• 2001

As a companion to an adiabatic version developed by Ryu and his coworkers, we have built an isothermal magnetohydrodynamic code for astrophysical flows. It is suited for the dynamical simulations of flows where cooling timescale is much shorter than dynamical timescale, as well as for turbulence and dynamo simulations in which detailed energetics are unimportant. Since a simple isothermal equation of state substitutes the energy conservation equation, the numerical schemes for isothermal flows are simpler (no contact discontinuity) than those for adiabatic flows and the resulting code is faster. Tests for shock tubes and Alfven wave decay have shown that our isothermal code has not only a good shock capturing ability, but also numerical dissipation smaller than its adiabatic analogue. As a real astrophysical application of the code, we have simulated the nonlinear three-dimensional evolution of the Parker instability. A factor of two enhancement in vertical column density has been achieved at most, and the main structures formed are sheet-like and aligned with the mean field direction. We conclude that the Parker instability alone is not a viable formation mechanism of the giant molecular clouds.

• Journal of The Korean Astronomical Society
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• v.34 no.4
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• pp.191-201
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• 2001

My contribution to these proceedings summarizes a general overview on High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. In the first part I will show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. I will review recent literature concerning the main properties of different special relativistic Riemann solvers, and discuss several 1D and 2D test problems which are commonly used to evaluate the performance of numerical methods in relativistic hydrodynamics. In the second part I will illustrate the use of HRSC methods in several astrophysical applications where special and general relativistic hydrodynamical processes play a crucial role.

• Communications for Statistical Applications and Methods
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• v.31 no.5
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• pp.535-556
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• 2024

Measuring stock market volatility and its determinants is critical for stock market participants, as volatility spillover effects affect corporate performance. This study adopted a novel approach to analysing and implementing GARCH-MIDAS modelling methods. The classical GARCH as a benchmark and the univariate GARCH-MIDAS framework are the GARCH family models whose forecasting outcomes are examined. The outcome of GARCH-MIDAS analyses suggests that inflation, interest rate, exchange rate, and oil price are significant determinants of the volatility of the Johannesburg Stock Market All Share Index. While for Nigeria, the volatility reacts significantly to the exchange rate and oil price. Furthermore, inflation, exchange rate, interest rate, and oil price significantly influence Ghanaian equity volatility, especially for the long-term volatility component. The significant shock of the oil price and exchange rate to volatility is present in all three markets using the generalized autoregressive conditional heteroscedastic-mixed data sampling (GARCH-MIDAS) framework. The GARCH-MIDAS, with a powerful fusion of the GARCH model's volatility-capturing capabilities and the MIDAS approach's ability to handle mixed-frequency data, predicts the volatility for all variables better than the traditional GARCH framework. Incorporating these two techniques provides an innovative and comprehensive approach to modelling stock returns, making it an extremely useful tool for researchers, financial analysts, and investors.

• Ocean Systems Engineering
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• v.12 no.3
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• pp.359-384
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• 2022

We develop in this work a new well-balanced preserving-positivity path-conservative central-upwind scheme for Saint-Venant-Exner (SVE) model. The SVE system (SVEs) under some considerations, is a nonconservative hyperbolic system of nonlinear partial differential equations. This model is widely used in coastal engineering to simulate the interaction of fluid flow with sediment beds. It is well known that SVEs requires a robust treatment of nonconservative terms. Some efficient numerical schemes have been proposed to overcome the difficulties related to these terms. However, the main drawbacks of these schemes are what follows: (i) Lack of robustness, (ii) Generation of non-physical diffusions, (iii) Presence of instabilities within numerical solutions. This collection of drawbacks weakens the efficiency of most numerical methods proposed in the literature. To overcome these drawbacks a reformulation of the central-upwind scheme for SVEs (CU-SVEs for short) in a path-conservative version is presented in this work. We first develop a finite-volume method of the first order and then extend it to the second order via the averaging essentially non oscillatory (AENO) framework. Our numerical approach is shown to be well-balanced positivity-preserving and shock-capturing. The resulting scheme could be seen as a predictor-corrector method. The accuracy and robustness of the proposed scheme are assessed through a carefully selected suite of tests.