Stiff Two-point Boundary Value Problems Research Articles

An efficient approach for solving stiff nonlinear boundary value problems

A new method for solving stiff two-point boundary value problems is described and compared to other known approaches using the Troesch’s problem as a test example. The method is based on the general idea of alternate approximation of either the unknown function or its inverse and has a genuine “immunity” towards numerical difficulties invoked by the rapid variation (stiffness) of the unknown solution. A c++ implementation of the proposed method is available at https://github.com/imathsoft/MathSoftDevelopment .

A SIMPLE APPROACH TO ANALYZE THE FULLY DEVELOPED TWO-PHASE MAGNETOCONVECTION TYPE FLOWS IN INCLINED PARALLEL-PLATE CHANNELS

A numerical approach is presented herein to analyze steady-state, fully developed MHDtype stratified two-fluid flow with heat transfer in inclined parallel-plate channels. The one dimensional model is extended to include several effects, such as buoyancy, moving plates, porous layers, inclined applied magnetic field, Joule and viscous heating and volumetric heat generation/absorption. Evaluation includes axial induced magnetic field, although its effect on the flow is negligible. The plates are maintained at constant temperatures, and the thermophysical properties are considered to be constant. The governing non-linear and coupled equations are solved through a well-established numerical routine to solve stiff two-point boundary value problems. The results are presented and validated for a large range of governing parameters and situations. Keywords   , ,, ,

Lobatto deferred correction for stiff two-point boundary value problems

An iterated deferred correction algorithm based on Lobatto Runge-Kutta formulae is developed for the efficient numerical solution of nonlinear stiff two-point boundary value problems. An analysis of the stability properties of general deferred correction schemes which are based on implicit Runge-Kutta methods is given and results which are analogous to those obtained for initial value problems are derived. A revised definition of symmetry is presented and this ensures that each deferred correction produces an optimal increase in order. Finally, some numerical results are given to demonstrate the superior performance of Lobatto formulae compared with mono-implicit formulae on stiff two-point boundary value problems.

Runge-Kutta methods for the solution of stiff two-point boundary value problems

One of the most popular approaches to the numerical solution of two-point boundary value problems is shooting. However this approach is often ineffective for singularly perturbed problems due to the possible presence of rapidly increasing modes which cannot be dealt with using an initial value solver. In this paper we survey the use of implicit Runge-Kutta methods for such problems. It is shown that certain classes of implicit Runge-Kutta formulae are both stable and accurate for singularly perturbed problems and an efficient implementation of these formulae, based on the use of deferred correction, is described.

Mesh selection for stiff two-point boundary value problems

This paper is concerned with the mesh selection algorithm of COLSYS, a well known collocation code for solving systems of boundary value problems. COLSYS was originally designed to solve non-stiff and mildly stiff problems only. In this paper we show that its performance for solving extremely stiff problems can be considerably improved by modifying its error estimation and mesh selection algorithms. Numerical examples indicate the superiority of the modified algorithm.

Automated allocation of mesh points for stiff two-point boundary value problems

The numerical solution of a stiff two-point boundary value problem is considered where a general procedure to automate the allocation of mesh points is obtained. This procedure requires negligible prerequisites to the problem. The automation is achieved by constructing a diffeomorphism ψ acting on the domain of the independent variable. This ψ smooths out the sharp corners in the solution while it leaves the domain boundary and the boundary conditions unchanged. The diffeomorphism is defined by differential equations which are coupled with the original equations to form an augmented system.

Mathematical evidence for flow-induced changes in myocardial oxygen consumption.

The objective of this investigation was to aid in the determination of the mechanism by which oxygen consumption changes in proportion to coronary perfusion pressure or coronary blood flow. A mathematical model of oxygen transport and consumption in the isolated-perfused heart was developed, based on data from an autoregulating, cell-free perfused, externally paced, isovolumic feline heart preparation. The model features the unique combination of Michaelis-Menten kinetics, and one-dimensional (axial) diffusion in radially well-mixed tissue. An adaptive finite-difference integration routine was used to solve the resulting third order stiff two-point boundary value problem. A simplex minimization was employed to determine the parameter values that minimized the squared difference between the model and the experimental data in terms of tissue PO2 distribution (histograms). Different cases of the model representing pressure-induced, flow-induced, and "magnified" flow effects were run. The flow-dependent oxygen consumption version of the model produced a histogram squared error 30% lower than the pressure-induced version and 5% lower than any other case. The model and a critical review of the literature indicate that a flow-related mechanism is responsible for this phenomenon. Evidence also demonstrates that the Michaelis-Menten kinetics constant is not constant for different oxygen tensions.

Numerical Methods for Stiff Two-Point Boundary Value Problems

We consider the two-point boundary value problem for stiff systems of ordinary differential equations. For systems that can be transformed to essentially diagonally dominant form with appropriate smoothness conditions, a priori estimates are obtained. Problems with turning points can be treated with this theory, and we discuss this in detail. We give robust difference approximations and present error estimates for these schemes. In particular we give a detailed description of how to transform a general system to essentially diagonally dominant form and then stretch the independent variable so that the system will satisfy the correct smoothness conditions. Numerical examples are presented for both linear and nonlinear problems.

Stability of Collocation at Gaussian Points

Symmetric Runge–Kutta schemes are particularly useful for solving stiff two-point boundary value problems. Such A-stable schemes perform well in many cases, but it is demonstrated that in some instances the stronger property of algebraic stability is required. A characterization of symmetric, algebraically stable Runge–Kutta schemes is given. The class of schemes thus defined turns out not to be very rich: The only collocation schemes in it are those based on Gauss points, and other schemes in the class do not seem to offer any advantage over collocation at Gaussian points.

A finite element method with a large mesh-width for a stiff two-point boundary value problem

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two-point boundary value problem on [−1, 1] ϵ u″ + pu′ = f, u(−1) = u(1) = 0, 0 < ϵ ∠ 1, (′ = d dx ) On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint. We discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh-points, the error depending on ϵ, the mesh-width and the degree of the piece-wise polynomials.

G5.3: Mathematical Modelling of a Steam Generator

A detailed nonlinear model of a once-through steam generator of BENSON® type is presented with special emphasis placed on accurate modelling of two-phase heat transfer. The nonlinear coupled partial differential equations were derived from first principles and thus permit predictions of the dynamic response of the unit already in the planning stage. The model equations, comprising a stiff two-point boundary value problem, are solved by means of an implicit Runge-Kutta-method featuring step-size control with prespecified numerical error. Using an optimized solution algorithm, computational requirements are kept modest. Comparison of simulation results to actual data shows good agreement and demonstrates the validity of the model. In addition, new insight into the physical phenomena occuring in the two-phase region is gained.

A Unified View of Some Methods for Stiff Two-Point Boundary Value Problems

The paper analyzes a number of known methods for the solution of two-point boundary value problems for ordinary differential equations; the methods considered have in common that they lead to a seq...

Galerkin solutions of stiff two-point boundary-value problems

Through the example of Conte (Ref. 6), the Galerkin procedure with a small number (N⩽6) of low-degree polynomial modes is illustrated as a computationally rapid and effective technique for solving extremely stiff linear two-point boundary-value problems. Numerical solutions are provided for eigenvalue spreads σ ranging from 20 through 106. They agree with the exact solution to at least 2N decimal places. The errors are insensitive to the eigenvalue spread. Comparisons are made with the continuation technique of Roberts and Shipman (Ref. 1), who did not succeed in solving this example for σ=√(36,000).