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.

Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation

Abstract: We analyze a class of numerical schemes for solving the HJB equation for stochastic control problems, which enters the framework of Markov chain approximations and generalizes the usual finite difference method. The latter is known to be monotonic, and hence valid, only if the scaled covariance matrix is dominant diagonal. We generalize this result by, given the set of neighboring points allowed to enter the scheme, showing how to compute effectively the class of covariance matrices that is consistent with this set of points. We perform this computation for several cases in dimensions 2, 3, and 4.

A meshless model for transient heat conduction in functionally graded materials

Abstract: A meshless numerical model is developed for analyzing transient heat conduction in non-homogeneous functionally graded materials (FGM), which has a continuously functionally graded thermal conductivity parameter First, the analog equation method is used to transform the original non-homogeneous problem into an equivalent homogeneous one at any given time so that a simpler fundamental solution can be employed to take the place of the one related to the original problem Next, the approximate particular and homogeneous solutions are constructed using radial basis functions and virtual boundary collocation method, respectively Finally, by enforcing satisfaction of the governing equation and boundary conditions at collocation points of the original problem, in which the time domain is discretized using the finite difference method, a linear algebraic system is obtained from which the unknown fictitious sources and interpolation coefficients can be determined Further, the temperature at any point can be easily computed using the results of fictitious sources and interpolation coefficients The accuracy of the proposed method is assessed through two numerical examples

Difference Solutions of the Shallow-Water Equation

Abstract: The quasi-linear partial differential equations known as the shallow-water equations describe the flow of water in open channels or over sloping planes (overland flow). Because no analytic solution exists for these equations, finite-difference methods must be used to obtain solutions. Formulation of finite-difference schemes involves consideration of the convergence of the finite-difference solutions to the true solution of the equation. A theoretical examination of the approximation of several difference operators, both implicit and explicit, is presented and the stability of these schemes is examined empirically for flow over a plane with critical depth downstream boundary condition and a zero inflow upstream boundary condition. A finite-difference scheme based on the method of characteristics was found to be satisfactory in many cases. Explicit methods were found to be not suitable for this problem except in some special cases.

Finite difference methods for two-point boundary value problems involving high order differential equations

Abstract: We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y (2n)+f(x,y)=0,y (2j)(a)=A 2j ,y (2j)(b)=B 2j ,j=0(1)n−1,n≧2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.