Hermite Interpolation Approach to High-Order Approximation of Heat Equations

Abstract


 
 
 It is usually desirable to approximate the solution of mathemati- cal problems with high-order of accuracy and preferably using com- pact stencils. This work presents an approach for deriving high-order compact discretization of heat equation with source term. The key contribution of this work is the use of Hermite polynomials to reduce second order spatial derivatives to lower order derivatives. This does not involve the use of the given equation, so it is universal. Then, Tay- lor expansion is used to obtain a compact scheme for first derivatives. This leads to a fourth-order approximation in space. Crank-Nicholson scheme is then applied to derive a fully discrete scheme. The result- ing scheme coincides with the fourth-order compact scheme, but our derivation follows a different philosophy which can be adapted for other equations and higher order accuracy. Two numerical experiments are provided to verify the fourth-order accuracy of the approach.
 
 

On Computational Efficiency of Nine Iterative Algorithms for Hammerstein Equations

“…The present work extends the work in [8] by investigating the performance of nine fixed point methods with regards to their computational costs on five nonlinear integral equations. In section 2, we present the nine discrete fixed point iteration schemes for solving (1)- (2).…”

“…This method is represented as if it has three operator evaluations but it actually has only two. This is why recent study [8] has found that the schemes of Argawal and coworkers and that of Ishikawa have approximately equal efficiency and even accuracy. Noor [9] proposed a three-step method with three operator evaluations, see also [10] for another such method.…”

“…It can be shown that H un i is a fourth-order approximation of T un(x) at xi ∈ Ω h , see [8]. From the definition of H un i , we obtain H g i as…”

“…Recently, there have been growing interest in using fixed point methods to solve integral equations, see [29,52,53,54,55,56] for example. In [8], three fixed point methods were compared with Newton method for solving nonlinear Fredholm equations.…”