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A numerical continuation method for parameter-dependent boundary value problems using bvpsuite 2.0
Abstract
The Matlab package bvpsuite 2.0 is a numerical collocation code for the approximation of solutions of a broad range of boundary value problems in ordinary differential equations. In this article, its newly implemented pathfollowing module with automated step-length control is presented. Two versions using the pseudo-arclength continuation method, allowing pathfollowing beyond turning points, were developed, both taking advantage of the existing features of bvpsuite 2.0 such as error estimation and mesh adaptation. The first version is based on the Gauss-Newton method. The second version is now contained in the package bvpsuite 2.0 and uses its built-in iterative method, the Fast Frozen Newton method. Their operating principles are presented and their performance is compared by means of the computation of some pathfollowing problems. Furthermore, the results of computations with bvpsuite 2.0 for a problem with path bifurcations are presented.
... The classes of the first and second order BVPs in singular ODEs have been extensively studied and various approaches to their numericals solution have been proposed, among them finite difference schemes and collocation methods, 1 see [18,19,26,38,39]. Collocation schemes proved especially robust and efficient and therefore, they have been used in many codes as basic solvers, see Fortran codes, COLSYS [6] and COLNEW [5,8], and Matlab codes MIRKDC [17], bvp4c [34,35], sbvp [3], and bvpsuite [4,22,24]. ...
... Collocation method and bvpsuite2.0In the scope of the code are BVPs for systems of implicit mixed order4 ODEs, ( , 1 , ..., , 1 ( ), ′ 1 ( ), ..., ( 1 ) 1 ( ), ..., ( ), ′ ( ), ..., ( ) ( )) = 0, (8) ( 1 , ..., , 1 ( 1 ), ..., ( 1 −1) 1 ( 1 ), ..., ( 1 ), ..., ( −1) ( 1 ), ..., 1 ( ), ..., ( 1 −1) 1 ...
... The codes colsys/colnew/colmod use Gauss-Legendre formulas of high order [2,5]. Also the code bvpsuite [3] is based on collocation and it is possible to choose as collocation points the zeros of the Legendre polynomial. In all the cited codes the adopted continuous extension is the collocation piecewise polynomial p which in this case is just C 0 smooth. ...
Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced,
where derivative information up to order R ≥ 1 at the spline breakpoints
is required and maximal convergence order can be proved. Relying on the B-spline
basis with possible multiple inner knots, a family of quasi-interpolating
splines with smoothness of order R is associated with each R ≥ 1, since there
is the possibility of using different local sequences of breakpoints to define each
QI spline coefficient. By using suitable finite differences approximations of the
necessary discrete derivative information, each QI spline in this family can be
associated also with a twin approximant belonging to the same spline space, but
requiring just function information at the breakpoints.
Among possible different applications of the introduced QI scheme, a smooth
continuous extension of the numerical solution of Gauss-Lobatto and Gauss-
Legendre Runge-Kutta methods is here considered. When R > 1, such extension
is based on the use of the variant of the QI scheme with derivative approximation
which preserves the approximation power of the original Runge-Kutta scheme.
Various analytical models that calculate the water flow either around or inside plant roots are available, but a combined analytical solution has not yet been derived. The classical solution of Landsberg and Fowkes for water flow within a root relates the second derivative of xylem water potential to the radial water influx term. This term can be linked to well-known steady state or steady rate-based solutions for computing soil water fluxes around roots. While neglecting lateral fluxes between local depletion zones around roots, we use this link to construct a system of continuous equations that combine root internal and external water flow that can be solved numerically for two boundary conditions (specified root collar water potential and zero distal influx) and one constraint (mean bulk matric flux potential). Furthermore, an iterative matrix solution for the stepwise analytical solution of homogeneous root segments is developed. Besides accounting for soil water flow iteratively, the intrinsic effect of variable axial conductance is accounted simultaneously. The reference and the iterative matrix solution are compared for different types of corn roots, soil textures and soil dryness states, which showed good correspondence. This also revealed the importance of accounting for variable axial conductance in more detail. The proposed reference solution can be used for the evaluation of different morphological and hydraulic designs of single or multiple parallel-connected roots operating in targeted soil environments. Some details of the iterative matrix solution may be adopted in analytical–numerical solutions of water flow in complex root systems.
The aim of this paper is to present the potentiality of the code HOFiD_bvp based on high order finite difference, following the approach of the boundary value methods and the upwind methods. Strategies, as variable stepsize and variable order, that helps to handle with solutions of singular perturbation problems, so as the continuation strategy applied for solving nonlinear problems, will be discussed. Now, the method is suitable to solve second order scalar boundary value problems, not only singularly perturbed with one and two parameters, but also problems with discontinuous source terms. Concerning that point some numerical tests will be taken in consideration in order to show the potentiality of the code.
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