Topic
Finite difference
About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.
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TL;DR: In this paper, Adomian's decomposition scheme is presented as an alternate method for solving the nonlinear Klein?Gordon equation, which is demonstrated by several examples and compared with existing collocation, finite difference and finite element techniques.
86 citations
01 Jun 1991
TL;DR: In this paper, the authors consider wave equations with damping in the boundary conditions and present techniques to ascertain the uniform preservation under approximation of exponential stability, and several schemes for which preservation can be guaranteed are analyzed.
Abstract: We consider wave equations with damping in the boundary conditions. Techniques to ascertain the uniform preservation under approximation of exponential stability are presented. Several schemes for which preservation can be guaranteed are analyzed. Numerical results that demonstrate the lack of stability under approximation for several popular schemes (including standard finite difference and finite element schemes) are given.
86 citations
01 Jul 1993
TL;DR: The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations and an application to flow around an airfoil is presented.
Abstract: The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.
86 citations
TL;DR: A numerical method for a generalized Black-Scholes equation, used for option pricing, based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique that efficiently treats the singularities of the non-smooth payoff function.
86 citations
TL;DR: The exactness and the feasibility of the interface conditions are investigated for the high-order multi-block computation on structured grid containing singular points and newly derived from the characteristic relations of the compressible Euler or Navier-Stokes equations are investigated.
Abstract: A structured grid with a body usually has a certain point where an abrupt change in the slope of grid line exists. The grid metrics are discontinuous at the point because of the discrepancy between the left- and the right-hand limits of the gradients, which leads to grid singularity. It may cause serious numerical oscillations especially when high-order finite difference schemes are applied to solving conservation-form governing equations in generalized coordinates. In this paper, it is handled by decomposing a computational domain into blocks along the singular lines and imposing interface conditions at the block interfaces for communication between the blocks. A set of high-order finite difference schemes is used in each block: central differences on the interior nodes and one-sided differences on the near-interface nodes. The differencing stencils do not cross the block interfaces and each block is isolated without the singularity,
which results in no oscillations. For the communication between the isolated blocks, the interface conditions are newly derived from the characteristic relations of the compressible Euler or Navier-Stokes equations. The exactness and the feasibility of the interface conditions are investigated for the high-order multi-block computation on structured grid containing singular points.
86 citations