A Higher-Order Scheme for Quasilinear Boundary Value Problems with Two Small Parameters

Vulanović, Relja

doi:10.1007/s006070170002

Cited by 71 publications

(34 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The use of L < ln N enables meshes with a greater density in the layers, which improves accuracy of numerical results. This is of practical importance only since, theoretically, any L behaves like ln N as N → ∞, see [21]. It should also be noted that the constant a is another parameter for controlling the mesh density in the layers; the smaller the value of a, the greater the density.…”

Section: Meshmentioning

confidence: 99%

“…It is easy to verify that the discriminant of the quadratic function ω is nonpositive if τ N The standard decomposition argument does not apply to G. Rather, it is combined with the technique from [21]. G is decomposed as G = A + K, where K = [k ij ] and…”

Section: N/2 Ons(l)mentioning

confidence: 99%

“…These exceptional points are the transition points, a couple of points adjacent to them, and perhaps a few points in the vicinity of x = 0 and x = 1. This approach has already been applied in [20,21] to some nonlinear problems with two small parameters, which are discretized by four-point third-order schemes.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems

Vulanović

2004

Comput. Methods Appl. Math.

Self Cite

View full text Add to dashboard Cite

-The discretization meshes of the Shishkin type are more suitable for highorder finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.2000 Mathematics Subject Classification: 65L10; 65L12; 65L50.

show abstract

Section: Meshmentioning

confidence: 99%

Section: N/2 Ons(l)mentioning

confidence: 99%

See 1 more Smart Citation

An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems

Vulanović

2004

Comput. Methods Appl. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Vulanović [17] considers finite difference methods in the case of µ = ε 1 2 +λ , λ > 0. More recently, parameter-uniform numerical methods for the steady-state version of (1.1) were examined by Linß and Roos [4], Roos and Uzelac [11] and O'Riordan et al [9].…”

mentioning

confidence: 99%

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

2006

View full text Add to dashboard Cite

Abstract. In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

show abstract

“…Numerical methods for one-dimensional singularly perturbed problems with two small parameters are considered in [3,4,[6][7][8][10][11][12], but on a Shishkin-type mesh. Vulanović [12] considered Shishkin and Bakhvalov meshes but assumed ε 2 = ε p+1/2 1 with p > 0.…”

Section: Introductionmentioning

confidence: 99%

Convection‐diffusion‐reaction problems on a B‐type mesh

Brdar

Zarin

2013

Proc Appl Math and Mech

View full text Add to dashboard Cite

One-dimensional singularly perturbed problems with two small parameters are considered. Numerical methods for such problems are discussed in several papers, but on a Shishkin-type mesh. The first optimal result of convergence in an energy norm on a Bakhvalov-type mesh for one-dimensional convection-diffusion problem was given by Roos in [9]. In this paper we analyze Galerkin finite element method on a Bakhvalov-type mesh for two-parameter convection-diffusion-reaction problems. In the interpolation error analysis, instead of the usual interpolation operator in the finite element space we use a quasiinterpolant with improved stability properties. We prove that the finite element method for these problems is uniformly convergent in the energy norm. Numerical results confirm our theoretical analysis and show first-order convergence rate.

show abstract

A Higher-Order Scheme for Quasilinear Boundary Value Problems with Two Small Parameters

Cited by 71 publications

References 13 publications

An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems

An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

Convection‐diffusion‐reaction problems on a B‐type mesh

Contact Info

Product

Resources

About