Osculating paraboloids of second and third order

Osculating paraboloids of second order of a surface have been discussed in classical affine differential geometry. We generalize this concept to cubic osculating paraboloids. This yields a visualization of the local properties of a given surface which depend on the derivatives of maximal order four.     

Two-step fourth order methods for linear ODEs of the second order

A family of two step difference schemes of the fourth order has been developed for linear ODEs of the second order. Stability properties for such schemes are discussed and results of numerical tests are given. It is shown how the proposed technique can be extended to non-linear ODEs of second order.     

Higher order difference methods for second order two-point boundary-value problems

Using methods of numerical integration, difference schemes of sixth order with off-step points have been obtained and applied to the second order differential equation, with and without mixed boundary conditions. Numerical results obtained by these methods have been compared with those obtained by using h⁴-extrapolation of the Numerov method. It is found that the sixth-order method based upon 4-point Lobatto quadrature with approximation 2 is more economical and accurate from the computational viewpoint than the existing sixth-order methods.     

Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations

In this paper, two Chebyshev-like third order methods free from second derivatives are considered and analyzed for systems of nonlinear equations. The methods can be obtained by having different approximations to the second derivatives present in the Chebyshev method. We study the local and third order convergence of the methods using the point of attraction theory. The computational aspects of the methods are also studied using some numerical experiments including an application to the Chandrasekhar integral equations in Radiative Transfer.     

Unconditionally stable methods for second order differential equations

We use the concept of order stars (see [1]) to prove and generalize a recent result of Dahlquist [2] on unconditionally stable linear multistep methods for second order differential equations. Furthermore a result of Lambert-Watson [3] is generalized to the multistage case. Finally we present unconditionally stable Nyström methods of order 2s (s=1,2, ...) and an unconditionally stable modification of Numerov's method. The starting point of this paper was a discussion with G. Wanner and S.P. Nørsett. The author is very grateful to them.     

Variational formulation and projectional methods for the second order transport equation

Herein the variational problem for a second-order boundary value problem for the neutron transport equation is formulated. The projectional methods solving the problem are examined. The approach is compared with that based on the original untransformed form of the neutron transport equation.     

Families of two-step fourth order P-stable methods for second order differential equations

This paper deals with a class of symmetric (hybrid) two-step fourth order P-stable methods for the numerical solution of special second order initial value problems. Such methods were proposed independently by Cash [1] and Chawla [3] and normally require three function evaluations per step. The purpose of this paper is to point out that there are some values of the (free) parameters available in the methods proposed which can reduce this work; we study two classes of such methods. The first is the class of ‘economical’ methods (see Definition 3.1) which reduce this work to two function evaluations per step, and the second is the class of ‘efficient’ methods (see Definition 3.2) which reduce this work with respect to implementation for nonlinear problems. We report numerical experiments to illustrate the order, acuracy and implementational aspects of these two classes of methods.     

Splitting methods for second‐order initial value problems

We consider implicit integration methods for the solution of stiff initial value problems for second-order differential equations of the special form y' = f(y). In implicit methods, we are faced with the problem of solving systems of implicit relations. This paper focuses on the construction and analysis of iterative solution methods which are effective in cases where the Jacobian of the right‐hand side of the differential equation can be split into a sum of matrices with a simple structure. These iterative methods consist of the modified Newton method and an iterative linear solver to deal with the linear Newton systems. The linear solver is based on the approximate factorization of the system matrix associated with the linear Newton systems. A number of convergence results are derived for the linear solver in the case where the Jacobian matrix can be split into commuting matrices. Such problems often arise in the spatial discretization of time‐dependent partial differential equations. Furthermore, the stability matrix and the order of accuracy of the integration process are derived in the case of a finite number of iterations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Unconditionally stable noumerov-type methods for second order differential equations

We report a modification of Noumerov's method which produces a family of unconditionally stable fourth order methods fory''=f(t, y).     

Inexact generalized Newton methods for second order C-differentiable optimization

In this paper we define second order C-differentiable functions and second order C-differential operators, describe their some properties and propose an inexact generalized Newton method to solve unconstrained optimization problems in which the objective function is not twice differentiable, but second order C-differentiable. We prove that the algorithm is linearly convergent or superlinearly convergent including the case of quadratic convergence depending on various conditions on the objective function and different values for the control parameter in the algorithm.     

Two-step fourth orderP-stable methods for second order differential equations

A family of symmetric (hybrid) two-step fourth order methods is derived fory'=f(x,y). We then show the existence of a sub-family of these methods which when applied toy'=– ² y, real, areP-stable. We also note that a general (order) symmetric two-step method isP-stable iff it is unconditionally stable.     

Two-step fourth orderP-stable methods for second order differential equations

A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory=f(x,y). We then show the existence of a sub-family of these methods which when applied toy=–k ² y,k real, areP-stable.     

Evaluation of Chebyshev pseudospectral methods for third order differential equations

When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and extreme eigenvalue already reaching O(N ⁵ for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N ⁴. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach. This revised version was published online in August 2006 with corrections to the Cover Date.     

The probabilistic solution of the third boundary value problem for second order elliptic equations

Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan-see [S-V]) the probabilistic solution of the boundary value problem

Reduction of fourth order ordinary differential equations to second and third order Lie linearizable forms

Meleshko presented a new method for reducing third order autonomous ordinary differential equations (ODEs) to Lie linearizable second order ODEs. We extended his work by reducing fourth order autonomous ODEs to second and third order linearizable ODEs and then applying the Ibragimov and Meleshko linearization test for the obtained ODEs. The application of the algorithm to several ODEs is also presented.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

Finite element methods for nonlinear elliptic systems of second order

This paper deals with the finite element displacement method for approximating isolated solutions of general quasilinear elliptic systems. Under minimal assumptions on the structure of the continuous problems it is shown that the discrete analogues also have locally unique solutions which converge with quasi-optimal rates in L₂ and L∞. The essential tools of the proof are a deformation argument and a technique using weighted L₂-norms.     

Univariate modified Fourier methods for second order boundary value problems

We develop and analyse a new spectral-Galerkin method for the numerical solution of linear, second order differential equations with homogeneous Neumann boundary conditions. The basis functions for this method are the eigenfunctions of the Laplace operator subject to these boundary conditions. Due to this property this method has a number of beneficial features, including an condition number and the availability of an optimal, diagonal preconditioner. This method offers a uniform convergence rate of , however we show that by the inclusion of an additional 2M basis functions, this figure can be increased to for any positive integer M.