Abstract
In this paper, the numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given. On the non-uniform grid of the uniformly distributed arc-length monitor function, the solution of the simple upwind scheme is obtained. It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters. Numerical experiments are carried out and the error analysis are confirmed.
Highlights
We consider a quasilinear singularly perturbed two-point boundary value problem: Tu ( x) := −εu′′( x) − b ( x,u ( x))′ + c ( x,u ( x)) = 0, x ∈ (0,1), (1)u= (0) u= (1) 0, where, 0 < ε 1 is a sufficiently small positive perturbation coefficient, b and c are sufficiently smooth functions, and for any x ∈[0,1],u ∈ R satisfies 0 < β ≤ bu (x,u) ≤ β *, 0 ≤ bx (x,u) ≤ C and 0 ≤ cu (x,u) ≤ γ *, 0 ≤ cx (x,u) ≤ C respectively where β, β *, γ * and C is a generic positive constant, independent of the perturbation parameter ε
The numerical solution and its error analysis of quasilinear singular perturbation two-point boundary value problems based on the principle of equidistribution are given
It is proved that the adaptive simple upwind scheme based on the principle of equidistribution has uniform convergence for small perturbation parameters
Summary
The problem was discretized using a simple upwind finite difference scheme on adapted meshes using grid equidistribution of monitor functions. In [6], Jugal and Srinivasan considered a linear problem and proved the uniform first-order convergence of the numerical solution on adapted meshes by equidistributing the arc-length monitor function. In [7], Linß, Roos and Vulanović considered the problem (1), which was discretized using a nonstandard upwinded first-order difference scheme on generalized Shishkin-type mesh. The problem is discretized using a simple upwind finite difference scheme on adapted meshes using grid equidistribution of the arc-length monitor function. We will prove the uniform first-order convergence and the numerical experiments verify our analysis
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