Uniformly convergent higher-order finite difference scheme for singularly perturbed parabolic problems with non-smooth data

Abstract

A uniformly convergent higher-order finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with non-smooth data. This scheme involves an average non-standard finite difference with the Richardson extrapolation method for space variables and second-order finite difference approximation for time direction on uniform meshes. The scheme is shown to be second-order convergent in both temporal and spatial directions. Further, the scheme is proven to be uniformly convergent and also confirmed by numerical experiments. Wide numerical experiments are conducted to support the theoretical results and to demonstrate its accuracy. Concisely, the present scheme is stable, convergent, and more accurate than existing methods in the literature.

A Numerical Method for Solving Singularly Perturbed Quasilinear Boundary Value Problems on Shishkin Mesh

“…Firstly, we consider the condition σ = l 2 . Taking l 2 ≤ α −1 ε ln N, we find h (1) = h (2) = h = lN −1 . Thus, it can be written that…”

“…Many different phonemena in science can be modeled by them. They emerge in electrochemistry [22], control theory [7], nuclear engineering [1], fluid dynamics [12] and plasma physics [6] (see, also the references therein).…”

“…Therefore, some important techniques have been introduced in the literature. These involve: finite difference methods [1,4,5,8,11,12,15], fitted mesh method [23], reproducing kernel method [7], factorization method [25], Galerkin method [18,26], differential transform method [9], variational iteration methods [2,10] and so on [6,16,19,20,27].…”

Accelerated nonstandard finite difference method for singularly perturbed Burger-Huxley equations

“…( 19 ), is convergent. A consistent and stable finite difference method is convergent by Lax's equivalence theorem [ 2 ]. For further convergence analysis, one can see the works provided in [ 5 , 11 , 13 , 14 ].…”

Second-order robust finite difference method for singularly perturbed Burgers' equation

“…Also, these problems have different properties and modeled problems depending on the dimensions, number of parameters involved, natures of the models, etc. For instance, some of them are singularly perturbed Burgers' equations, Burger-Huxley equations, Burger-Fisher equations, and so on, see the reference in [1,2,3,4,5,6,7,8,9]. Singularly perturbed Burgers' equation first introduced by Bateman in 1915 [9] is used in the modeling of the motion of the viscous fluid.…”