Accelerated convergence of Jameson's finite-volume Euler scheme using van der Houwen integrators

An efficient method for obtaining converged solutions to the time-dependent Euler equations has recently been proposed by Jameson. The convergence of the method is stabilised by using a four-stage Runge-Kutta scheme. In this paper the convergence is assessed of multistage algorithms which are of Runge-Kutta type, and are stable for larger time steps. This is done by means of numerical experiments using a coarse mesh. A six-stage algorithm is found to be best. It gives answers 15–20% more quickly than the standard method. No overheads in terms of increased storage are involved, and the results are indistinguishable at plotting accuracy from those obtained with the standard method.     

On the convergence of Wilson's nonconforming element for solving the elastic problems

A nonconforming finite element, Wilson's element, for solving the elastic problem is mathematically studied. This element passes the patch-test. The errors on the stresses and displacements are shown to be asymptotically of order $\text{h}$ and $\text{h}{}^2$, respectively, where $\text{h}$ is the supremum of the lengths of the sides of the elements.     

An integral equation method for the numerical solution of Laplace's equation without Green's function

The common way to establish an integral equation for the solution of Laplace's equation uses the Green's function of the given equation. It will be shown in this paper that an integral equation can also be constructed by using a particular solution of the Laplace equation as the Kernel of the integral equation.     

Hoare's logic and Peano's arithmetic

We develop the proof theory of Hoare's logic for the partial correctness of while- programs applied to arithmetic as it is defined by Peano's axioms. By representing the strongest postcondition calculus in Peano arithmetic PA, we are able to show that Hoare's logic over PA is equivalent to PA itself.     

Solving parabolic PDE's by a decomposition method

The paper deals with the application of the hybrid method of decomposition for the solution of two-dimensional parabolic partial differential equations with constant coefficients. The convergence and accuracy of the CSDT method are analysed, deriving the form of the relationship between ‘eigenvalues’ and the time step. Theoretical problems of this method of decomposition are mentioned. Also, the expression for computing the coefficients for various types of boundary problems has been derived. Practical applications of the method are presented. The cubic spline function interpolation is used for the continuous signal reproduction in the hybrid system. Some results obtained are presented at the end.     

A semi-local convergence theorem for a robust revised Newton’s method

It is well known that Newton’s iteration will abort due to the overflow if the derivative of the function at an iterate is singular or almost singular. In this paper, we study a robust revised Newton’s method for solving nonlinear equations, which can be carried out with a starting point with a degenerate derivative at an iterative step. It is proved that the method is convergent under the conditions of the Newton–Kantorovich theorem, which implies a larger convergence domain of the method. We also show that our method inherits the fast convergence of Newton’s method. Numerical experiments are performed to show the robustness of the proposed method in comparison with the standard Newton’s method.