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In this paper the technique of subtracting out singularities is used to derive explicit and implicit product Euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of Volterra integral equations with a weakly singular kernel. The convergence proofs of the numerical schemes are presented; these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global Lipschitz condition.

This paper analyses hopscotch algorithms when used to solve elliptic partial differential equations. A comparison with standard methods is made for the model problem.

Hopscotch, a fast finite difference technique, is used to solve parabolic and elliptic equations in two space dimensions with a mixed derivative. The method is compared numerically with existing alternating direction implicit (A.D.I.) and locally one dimensional (L.O.D.) methods for simple problems.Douglas and Gunn's A.D.I. method is both simplified and improved by reformulating it as a hopscotch method.

The monotonicity of a sequence arising in the convergence proof of a product integration scheme for a Volterra integro-differential equation is demonstrated.