Anisotropic mesh adaptation for numerical solution of boundary value problems

We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

The numerical solution of unstable two point boundary value problems

A number of workers have tried to solve, numerically, unstable two point boundary value problems. Multiple Shooting and Continuation Methods have been used very successfully for these problems, but each has weaknesses; for particularly unstable problems their success may be partial. In this paper we develop an algorithm that attempts to solve these problems in a routine manner.The algorithm uses a combination of Multiple Shooting and Range Extension in such a way that the advantages of both are maintained while the effects of their disadvantages are reduced considerably. The success of the algorithm is demonstrated on some particularly unstable problems.     

High order methods for the numerical solution of two-point boundary value problems

In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a one-off formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.     

Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline

In this paper, we present B-spline method for numerically solving singular two-point boundary value problems for certain ordinary differential equation having singular coefficients.These problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry. To remove the singularity, we first use Chebyshev economizition in the vicinity of the singular point and the boundary condition at a point x=δ (in the vicinity of the singularity) is derived. The resulting regular BVP is then efficiently treated by employing B-spline for finding the numerical solution. Some examples have been included and comparison of the numerical results made with other methods.     

A Taylor series method for the numerical solution of two-point boundary value problems

Summary For the numerical solution of two-point boundary value problems a shooting algorithm based on a Taylor series method is developed. Series coefficients are generated automatically by recurrence formulas. The performance of the algorithm is demonstrated by solving six problems arising in nonlinear shell theory, chemistry and superconductivity.     

A high order method for the numerical solution of two-point boundary value problems

A one step finite difference scheme of order 4 for the numerical solution of the general two-point boundary value problemy=f(t,y),a t b, withg(y(a),y(b))=0 is presented. The global discretization error of the scheme is shown, in sufficiently smooth cases, to have an asymptotic expansion containing even powers of the mesh size only. This justifies the use of Richardson extrapolation (or deferred correction) to obtain high orders of accuracy. A theoretical examination of the new scheme for large systems of equations shows that for a given mesh size it generally requires about twice as much work as the Keller box scheme. However, the expectation of higher accuracy usually justifies this extra computational effort. Some numerical results are given which confirm these expectations and show that the new scheme can be generally competitive with the box scheme.     

Time step selection for the numerical solution of boundary value problems for parabolic equations

An algorithm is proposed for selecting a time step for the numerical solution of boundary value problems for parabolic equations. The solution is found by applying unconditionally stable implicit schemes, while the time step is selected using the solution produced by an explicit scheme. Explicit computational formulas are based on truncation error estimation at a new time level. Numerical results for a model parabolic boundary value problem are presented, which demonstrate the performance of the time step selection algorithm.     

Continuation and shooting methods for boundary value problems of Bernstein type

Fixed point continuation methods and shooting methods are combined to produce an effective numerical procedure for solving boundary value problems for nonlinear ordinary differential equations. Typical numerical solution schemes involve an iteration procedure. Continuation methods systematically generate good initial guesses and, when combined with a shooting method and an appropriate update procedure, give systematic means for the numerical solution of nonlinear boundary value problems. This paper concentrates on problems of Bernstein type, which arise naturally in the calculus of variations and in steady-state heat conduction.     

Newton's method with mesh refinements for numerical solution of nonlinear two-point boundary value problems

Summary The object of this paper is the numerical solution of nonlinear two-point boundary value problems by Newton's method applied to the discretized problem on successively refined grids. The first part consists of a theoretical development of a phenomenon that often occurs in practice, namely, that the number of iterations for Newton's method to converge to within a fixed tolerance and for a fixed starting vector is essentially independent of the mesh size. The second part develops a process based on these results for determining an efficient mesh refinement strategy. Numerical results are also provided.This work was supported by N.S.F. grant GJ42626     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

On the use of splines for the numerical solution of nonlinear two-point boundary value problems

Cubic splines on splines and quintic spline interpolations are used to approximate the derivative terms in a highly accurate scheme for the numerical solution of two-point boundary value problems. The storage requirement is essentially the same as for the usual trapezoidal rule but the local accuracy is improved fromO(h ³) to eitherO(h ⁶) orO(h ⁷), whereh is the net size. The use of splines leads to solutions that reflect the smoothness of the slopes of the differential equations.     

Modified hierarchy basis for solving singular boundary value problems

In this paper, we develop an efficient preconditioning method on the basis of the modified hierarchy basis for solving the singular boundary value problem by the Galerkin method. After applying the preconditioning method, we show that the condition number of the linear system arising from the Galerkin method is uniformly bounded. In particular, the condition number of the preconditioned system will be bounded by 2 for the case q(x)=0 (see Eq. (1) in the paper). Numerical results are presented to confirm our theoretical results.     

A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

A numerical method for solving singular boundary value problems

Summary A numerical method is treated for solving singular boundary value problems with solutions that can be represented as series expansions on a subinterval near the singularity. A regular boundary value problem is derived on the remaining interval, for which a difference method is used. Convergence theorems are given for general schemes and for schemes of positive type for second order equations.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.