A finite differences MATLAB code for the numerical solution of second order singular perturbation problems

Abstract

We show the main features of the MATLAB code HOFiD_UP for solving second order singular perturbation problems. The code is based on high order finite differences, in particular on the generalized upwind method. Within its simplicity, it uses order variation and continuation for solving any difficult nonlinear scalar problem. Several numerical tests on linear and nonlinear problems are considered. The best performances are reported on problems with perturbation parameters near the machine precision, where most of the codes for two-point BVPs fail

Asymptotical computations for a model of flow in saturated porous media

“…The idea of discretizing each derivative of the second order continuous problem by means of finite difference schemes has been largely considered in the past, in particular in the numerical solution of partial differential equations using schemes of low order (and with small stencils). In [4] and later in [1,3,5] it was proposed to apply high order finite difference schemes for the solution of boundary value problems for ordinary differential equations by considering different formulae with the same order in the initial and final points of the grid, following the idea inherited by boundary value methods [7]. One major advantage of this approach is associated with the fact that the vector of unknowns contains only the solution of the problem.…”

Near critical, self-similar, blow-up solutions of the generalised Korteweg–de Vries equation: Asymptotics and computations

“…The order of the method is then chosen depending on the tolerances and for nonlinear problems the continuation strategy is applied. Experimental results show that, in general, the grids from HoFiD_UP are coarser when compared with other available software [29]. The efficiency and robustness of the above schemes has been illustrated in the context of singular eigenvalue Sturm-Liouville problems [32], the porous medium equation [33], and multi-parameter BVPs with singular points [34].…”

“…HoFiD_UP [29] . This solver is based on finite difference schemes whose order is even and varies from 4 to 8, and can solve k-th order ODE systems including those of the form (93), subject to general nonlinear boundary conditions.…”

BVPs Codes for Solving Optimal Control Problems

“…The code HOFiD_bvp [20] is based on high-order finite difference schemes (HOFiD) of order four, six, eight and ten, and an upwind method. Each derivative in the high-order boundary value problem is approximated directly by these schemes, hence it is not required any transformation of the problem in a system of first-order differential equations.…”

“…The Matlab environment allows the use of two functions, named bvp4c [16] and bvp5c [17], for solving BVPs. Other interesting codes that are usable in Matlab are bvptwp [18], TOM [19], HOFiD_bvp [20] and bvpSuite2.0 [21], based on the code sbvp [22] for the solution of singular problems. The code bvpSuite2.0 could be used also for singular BVPs and differential algebraic problems of index 1.…”