Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
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TL;DR: First-order consistency, unconditional stability, and first-order convergence of the method are proven using a novel shifted version of the classical Grunwald finite difference approximation for the fractional derivatives.
162 citations
TL;DR: A semi-Lagrangian method for solving the HJB equation for a typical gas storage valuation problem is presented and is able to handle a wide class of spot price models that exhibit mean-reverting seasonality dynamics and price jumps.
Abstract: The valuation of a gas storage facility is characterized as a stochastic control problem, resulting in a Hamilton-Jacobi-Bellman (HJB) equation. In this paper, we present a semi-Lagrangian method for solving the HJB equation for a typical gas storage valuation problem. The method is able to handle a wide class of spot price models that exhibit mean-reverting seasonality dynamics and price jumps. We develop fully implicit and Crank-Nicolson timestepping schemes based on a semi-Lagrangian approach and prove the convergence of fully implicit timestepping to the viscosity solution of a modified HJB equation posed on a bounded domain, provided that a strong comparison result holds. The semi-Lagrangian approach avoids Policy-type iterations required by an implicit finite difference method without requiring additional cost. Numerical experiments are presented for several variants of the basic scheme.
161 citations
TL;DR: FDTD equations that allow us to use a nonuniform grid are derived and this grid gives a better accuracy to CPU–resource ratio in a number of circumstances, and tilted and curved boundaries can be described more easily.
Abstract: The finite‐difference time‐domain (FDTD) approximation can be used to solve acoustical field problems numerically. Mainly because it is a time‐domain method, it has some specific advantages. The basic formulation of the FDTD method uses an analytical grid for the discretization of an unknown field. This is a major disadvantage. In this paper, FDTD equations that allow us to use a nonuniform grid are derived. With this grid, tilted and curved boundaries can be described more easily. This gives a better accuracy to CPU–resource ratio in a number of circumstances. The paper focuses on the new formulation and its accuracy. The problem of automatically generating the mesh in a general situation is not addressed. Simulations using quasi‐Cartesian grids are compared to Cartesian grid results.
161 citations
TL;DR: In this paper, the effects of seven types of influential factors such as the Rayleigh and Lewis numbers, the buoyancy-ratio parameter, the Brownian motion parameter, thermophoresis parameter, thermal conductivity ratio, and solid walls thickness on the fluid flow and heat transfer have been determined.
161 citations
TL;DR: In this paper, an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations is presented.
Abstract: The Osher algorithm for solving the Euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations. A first-order accurate version of the numerical method was derived by Osher circa 1980 for the one-dimensional non-isentropic Euler equations in Cartesian coordinates. In this paper, the extension of the scheme to arbitrary two-dimensional geometries is explained. Results are then presented for several example problems in one and two dimensions. Future work will include extension of the method to second-order accuracy and the development of implicit time differencing for the Osher algorithm.
160 citations