A fourth-order spline method for singular two-point boundary-value problems

AbstractThis paper describes two methods for the solution of (weakly) singular two-point boundary-value problems: Consider the uniform mesh xi = ih, h = 1/N, i = 0(1)N. Define the linear functionals Li(y) = y(xi) and Mi(y) = (x−α(xαy′)′\xv;x=xi. In both these methods a piecewise ‘spline’ solution is obtained in the form s(x) = si(x), x\wE; [xi−1, xi], i = 1(1)N, where in each subinterval si(x) is in the linear span of a certain set of (non-polynomial) basis functions in the representation of the solution y(x) of the two-point boundary value problem and satisfies the interpolation conditions: Li−1(s) = Li−1(y), Li(y), Mi−1(s) = Mi−1(y), Mi(s) = Mi(y). By construction s and x−α(xαs′)′ \wE; C[0,1]. Conditions of continuity are derived to ensure that xαs′ \wE; C[0, 1]. It follows that the unknown parameters yi and Mi(y), i = 1(1)N − 1, must satisfy conditions of the form: The first method consists in replacing Mi(y) by fnof(xi, yi) and solving (*) to obtain the values yi; this method is generalization of the idea of Bickley [2] for the case of (weakly) singular two-point boundary-value problems and provides order h2 uniformly convergent approximations over [0, 1]. As a modification of the above method, in the second method we generate the solution yi at the nodal points by adapting the fourth-order method of Chawla [3] and then use the conditions of continuity (*) to obtain the corresponding smoothed approximations for Mi(y) needed for the construction of the spline solution. We show that the resulting new spline method provides order h4 uniformly convergent approximations over [0, 1]. The second-order and the fourth-order methods are illustrated computationally

High-order numerical solutions using cubic splines

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes

Convergence of a Boundary Integral Method for Water Waves

We prove nonlinear stability and convergence of certain boundary integral methods for time-dependent water waves in a two-dimensional, inviscid, irrotational, incompressible fluid, with or without surface tension. The methods are convergent as long as the underlying solution remains fairly regular (and a sign condition holds in the case without surface tension). Thus, numerical instabilities are ruled out even in a fully nonlinear regime. The analysis is based on delicate energy estimates, following a framework previously developed in the continuous case [Beale, Hou, and Lowengrub, Comm. Pure Appl. Math., 46 (1993), pp. 1269–1301]. No analyticity assumption is made for the physical solution. Our study indicates that the numerical methods must satisfy certain compatibility conditions in order to be stable. Violation of these conditions will lead to numerical instabilities. A breaking wave is calculated as an illustration