• The Bulletin of The Korean Astronomical Society
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• v.44 no.1
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• pp.46.1-46.1
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• 2019

We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James's method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green's function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the {\tt Athena++} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.

• Environmental and Resource Economics Review
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• v.19 no.3
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• pp.597-632
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• 2010

There is general agreement that global warming is occurring and that the main contributor to this probably is the buildup of green house gasses, GHG, in the atmosphere. Two main contributors are the utilization of fossil fuels and the deforestation of many regions of the world. The burning of fossil fuels increases atmospheric carbon while the burning of fuel-wood reducing fossil fuel consumption along with its forest source maintain an atmospheric carbon level. The standing timber in the forests is a carbon sink, as are wood buildings and structures, and fossil fuel in the ground. This paper is designed to examine a number of current issues related to mitigating the global warming problem through forestry. For this purpose, we develop a modeling approach by integrating timber market, fossil fuel market and carbon cycling model. We use discrete time optimal control theory to identify optimal time paths, the laws of motion, and stationary stats solutions of endogenous variables in the model. On the basis of these results, we identify the optimal amounts of subsidies to be provided or taxes to be imposed by the regulatory agency to mitigate atmospheric carbon accumulation. We also present a numerical example to help illustrate the characteristics of variables in the model when the social cost for atmospheric carbon incrementally shifts upward. A surprising result is that the social cost function for atmospheric carbon has a very smaller impact on the optimal rotation period than previous literature suggested.