High-order finite difference schemes for the solution of second-order BVPs

High Order Finite Difference Schemes for the Solution of Elliptic PDEs

“…−div(M (|∇u|)∇u) = λ, (x, y) ∈ Ω, u(x, y) = g (x, y) o n∂Ω (1) where Ω = [x 0 , x f ] × [y 0 , y f ], λ ∈ IR, | · | stands for the Euclidean norm in IR 2 , div is the divergence operator and M (·) is sufficiently regular in its argument. Problems of this kind describe, for example, mathematical models for elastomers and soft tissues (see e.g.…”

“…In this paper, we propose to solve (1) by stable finite difference schemes of high order on a regular domain with uniform meshgrid. The main idea is the application in more spatial dimensions of the new classes of Boundary Value Methods (BVMs) introduced in [1] to solve twopoint BVPs for second order ODEs. For this reason, in the first part of the paper we report on these formulae and their main properties.…”

“…We solve nonlinear elliptic PDEs by stable finite difference schemes of high order on a uniform meshgrid. These schemes have been introduced in [1] in the class of Boundary Value Methods (BVMs) to solve two-point Boundary Value Problems (BVPs) for second order ODEs and are high order generalizations of classical finite difference schemes for the first and second derivatives. Numerical results for a minimal surface problem and for the Gent model in nonlinear elasticity are presented.…”

Singular Perturbation Problems

“…An alternative approach would be for finite differences to approximate each derivative separately and these would be cheaper when they are directly applied to (1)- (2). In [1] some new methods in this class have been proposed that combine generalizations of the central differences to approximate the second derivative with generalization of forward/backward/central differences for the first derivative. Following the idea inherited from Boundary Value Methods (see [6] for a review on the approach) the main feature of this approach is that of combining one main formula with additional initial and final ones.…”

“…Codes that can be used to solve (1)-(2) are TWPBVP [11] and its variants [8], MIRKDC and its new implementation BVP SOLVER [19], COLSYS [4] and COLNEW [5] (see also [10]), and the MATLAB codes TOM [16] and BVP4c [18]. Most of the above numerical methods can only be applied to first-order equations and hence they require a rewriting of the original higher order problem (1). An alternative approach would be for finite differences to approximate each derivative separately and these would be cheaper when they are directly applied to (1)- (2).…”